THE STRUCTURE OF CROSSED PRODUCTS OF IRRATIONAL ROTATION ALGEBRAS BY FINITE SUBGROUPS OF SL_{2}(ℤ)

SIEGFRIED ECHTERHOFF, WOLFGANG LÜCK, N. CHRISTOPHER PHILLIPS, AND SAMUEL WALTERS

Abstract. Let F ⊆ SL_{2}(ℤ) be a finite subgroup (necessarily isomorphic to one of ℤ_{2}, ℤ_{3}, ℤ_{4}, or ℤ_{6}), and let F act on the irrational rotational algebra A_{θ} via the restriction of the canonical action of SL_{2}(ℤ). Then the crossed product A_{θ} ⋊_{α} F and the fixed point algebra A_{θ}^{F} are AF algebras. The same is true for the crossed product and fixed point algebra of the flip action of ℤ_{2} on any simple d-dimensional noncommutative torus A_{Θ}. Along the way, we prove a number of general results which should have useful applications in other situations.

0. Introduction and statement of the main results

For θ ∈ ℝ let A_{θ} be the rotation algebra, which is the universal C*-algebra generated by unitaries u and v satisfying vu = exp(2πiθ)uv. The group SL_{2}(ℤ) acts on A_{θ} by sending the matrix

n = ( \begin{matrix}n_{1,1} & n_{1,2} \\ n_{2,1} & n_{2,2} \end{matrix} )

to the automorphism determined by

(0.1) α_{n}(u) = exp(πin_{1,1}n_{2,1}θ)u^{n_{1,1}}v^{n_{2,1}} and α_{n}(v) = exp(πin_{1,2}n_{2,2}θ)u^{n_{1,2}}v^{n_{2,2}}.

In this paper we are concerned with the structure of the crossed products A_{θ} ⋊_{α} F for the finite subgroups F ⊆ SL_{2}(ℤ), which are, up to conjugacy, the groups F = ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6} (where ℤ_{m} stands for the cyclic group of order m) generated by the matrices

( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) (for ℤ_{2}), ( \begin{matrix}−1 & −1 \\ 1 & 0 \end{matrix} ) (for ℤ_{3}),

( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) (for ℤ_{4}),  and  ( \begin{matrix}0 & −1 \\ 1 & 1 \end{matrix} ) (for ℤ_{6}). (0.2)

We refer to Proposition 21 of [32] for a proof that one obtains essentially all interesting actions of the cyclic groups ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6} on A_{θ} by restricting the action α of SL_{2}(ℤ) to these subgroups. The main results of this paper culminate in a proof of the following theorem:

Date: 26 September 2006.

2000 Mathematics Subject Classification. Primary 19K14, 46L35, 46L55, 46L80; Secondary 18F25, 19K99, 46L40.

Research of the first and second authors partially supported by the Deutsche Forschungsgemeinschaft (SFB 478). Research of the third author partially supported by NSF grants DMS 0070776 and DMS 0302401.

Theorem 0.1 (Theorems 4.9, 6.3, and 6.4). Let F be any of the finite subgroups ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6} ⊆ SL_{2}(ℤ) with generators given as above and let θ ∈ ℝ ∖ ℚ. Then the crossed product A_{θ} ⋊_{α} F is an AF algebra. For all θ ∈ ℝ we have

K_{0}(A_{θ} ⋊_{α} ℤ_{2}) ≅ ℤ^{6}, K_{0}(A_{θ} ⋊_{α} ℤ_{3}) ≅ ℤ^{8},
K_{0}(A_{θ} ⋊_{α} ℤ_{4}) ≅ ℤ^{9}, and K_{0}(A_{θ} ⋊_{α} ℤ_{6}) ≅ ℤ^{10}.

For F = ℤ_{k} for k = 2, 3, 4, 6, the image of K_{0}(A_{θ} ⋊_{α} F) under the canonical tracial state on A_{θ}⋊_{α}F (which is unique) is equal to \frac{1}{k} (ℤ+θℤ). As a consequence, A_{θ}⋊_{α}ℤ_{k} is isomorphic to A_{θ′} ⋊_{α} ℤ_{l} if and only if k = l and θ′ = ±θ mod ℤ.

If θ ∈ ℝ∖ℚ then the fixed point algebra A_{θ}^{F} is Morita equivalent to A_{θ}⋊_{α}F. (This follows from the Proposition in [66].) Thus, as a direct corollary of Theorem 0.1 we get:

Corollary 0.2 (Corollary 6.5). Let α : F → Aut(A_{θ}) be as above with F = ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6}. Then the fixed point algebras A_{θ}^{F} are AF for all θ ∈ ℝ ∖ ℚ.

The proof of Theorem 0.1 has three independent steps:

(1) Computation of the K-theory of the crossed product.

(2) Proof that the crossed product satisfies the Universal Coefficient Theorem.

(3) Proof that the action of the group has the tracial Rokhlin property of [59]. Given these steps, one uses Theorem 2.6 of [59] to prove that the crossed product has tracial rank zero in the sense of [39, 40], and then Huaxin Lin’s classification theorem (Theorem 5.2 of [41]) to conclude that the crossed product is AF.

Our results have been motivated by previous studies of the algebras A_{θ} ⋊_{α} F and A_{θ}^{F} by several authors. See, for example, [7, 9, 26, 27, 28, 29, 30, 31, 32, 38, 62, 72, 73, 74]. The most was known about the crossed product by the flip: its (unordered) K-theory has been computed in [38], and the crossed product has been proved to be an AF algebra in [9]. The next best understood case is the case of ℤ_{4}, which has been intensively studied in a series of papers culminating in [74]. It is proved there that for most irrational θ (that is, a dense G_{δ}-set of them) the crossed product A_{θ} ⋊_{α} ℤ_{4} has tracial rank zero in the sense of [39], and that for most of those values of θ, it is an AF algebra. Step (1) is done for most θ in [73], Step (2) is done for all θ in [74], and, in place of Step (3), a direct proof is given in [74] that the crossed product has tracial rank zero for most θ. That A_{θ} ⋊_{α} ℤ_{4} is an AF algebra for most θ then follows from Lin’s classification theorem.

Using completely different methods, we prove in this paper several general results which will imply, as special cases, the three steps described above for all θ ∈ ℝ ∖ ℚ and for all choices for F = ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6}. For the computation of the K-theory (Step (1) above), we observe that the crossed products A_{θ} ⋊_{α} F can be realized as (reduced) twisted group algebras C_{r}^{∗}(ℤ^{2} ⋊ F, ω̃_{θ}), where ℤ^{2} ⋊ F is the semidirect product by the obvious action of F on ℤ^{2} and ω̃_{θ} ∈ Z^{2}(ℤ^{2} ⋊F, 𝕋) is a suitable circle valued 2-cocycle on ℤ^{2} ⋊ F. Since ω̃_{θ} is homotopic to the trivial cocycle in a sense explained below, the following result will imply that the K-groups of A_{θ} ⋊_{α} F are isomorphic to the K-groups of the untwisted group algebra C_{r}^{∗}(ℤ^{2}⋊F) ≅ C(𝕋^{2})⋊F, which we compute (together with an explicit basis) by using some general methods from [18].

CROSSED PRODUCTS OF IRRATIONAL ROTATION ALGEBRAS BY FINITE GROUPS 3

Theorem 0.3 (Theorem 1.9). Suppose that ω_{0}, ω_{1} ∈ Z^{2}(G, 𝕋) are homotopic Borel 2-cocycles on the second countable locally compact group G, that is, there exists a Borel 2-cocycle Ω ∈ Z^{2}(G, C([0, 1], 𝕋)) such that ω_{j} = Ω( · , ·)(j) for j = 0, 1. Suppose further that G satisfies the Baum-Connes conjecture with coefficients. (If G is amenable or a-T-menable, this is automatic by Theorem 1.1 of [33].) Then K_{∗}(C_{r}^{∗}(G, ω_{0})) ≅ K_{∗}(C_{r}^{∗}(G, ω_{1})).

Indeed, using the homotopy Ω ∈ Z^{2}(G, C([0, 1], 𝕋)) one can construct a C*- algebra C_{r}^{∗}(G, Ω) which is a continuous field of C*-algebras over [0, 1] with fibers isomorphic to the twisted group algebras C_{r}^{∗}(G, ω_{θ}), with ω_{θ} = Ω( · , ·)(θ) for θ ∈ [0, 1]. We show, under the conditions on G in Theorem 0.3 (in fact, slightly more generally), that the quotient maps q_{θ}: C_{r}^{∗}(G, Ω) → C_{r}^{∗}(G, ω_{θ}) induce isomorphisms on K-theory. This special structure of the isomorphism will be used for our description of an explicit basis of A_{θ} ⋊_{α} F: we extend the basis of K_{0}(C_{r}^{∗}(ℤ^{2} ⋊ F)) to a basis of K_{0}(C_{r}^{∗}(ℤ^{2} ⋊ F, Ω)) and then evaluate at the other fibers.

The K_{0}-groups of the crossed products of the irrational rotation algebras by the actions of ℤ_{2}, ℤ_{3}, ℤ_{4}, and ℤ_{6} have been computed independently in [62], using completely different methods. These methods give no information about the K_{1}- groups, and therefore do not suffice for the proof that the crossed products are AF algebras. We should also mention that Georges Skandalis pointed out to us an alternative way to compute the K-theory of the crossed products A_{θ} ⋊_{α} F, also using the Baum-Connes conjecture. However, the method described above seems best suited for computing an explicit basis for K_{0}(A_{θ} ⋊_{α} F).

Step (2) (the Universal Coefficient Theorem) will be obtained from the Baum-Connes conjecture by slightly extending ideas first used in Section 10.2 of [71] and Lemma 5.4 of [15].

Step (3) (the tracial Rokhlin property) is obtained via Theorem 5.5, according to which an action of a finite group on a simple separable unital nuclear C*-algebra with unique tracial state has the tracial Rokhlin property if and only if the induced action is outer on the type II_{1} factor obtained from the trace using the Gelfand-Naimark-Segal construction. Along the way, we give a characterization of the tracial Rokhlin property in terms of trace norms which does not require uniqueness of the tracial state.

The methods developed here can also be applied to the crossed products of all simple d-dimensional noncommutative tori A_{Θ}, for d ≥ 2, by the flip action of ℤ_{2}, which sends the unitary generators u_{1}, u_{2}, …, u_{d} to their adjoints u_{1}^{∗}, u_{2}^{∗}, …, u_{d}^{∗}. Using the general methods explained above together the main result of [60], which shows that every simple higher dimensional d-torus is an AT algebra, we prove:

Theorem 0.4 (Corollary 1.14, Theorem 6.6, and Corollary 6.7). Let A_{Θ} be the noncommutative d-torus corresponding to a real d × d skew symmetric matrix Θ. Let α: ℤ_{2} → Aut(A_{Θ}) denote the flip action. Then

K_{0}(A_{Θ} ⋊_{α} ℤ_{2}) ≅ ℤ^{3·2^{d−1}} and K_{1}(A_{Θ} ⋊_{α} ℤ_{2}) = {0}.

If A_{Θ} is simple, then A_{Θ} ⋊_{α} ℤ_{2} and the fixed point algebra A_{Θ}^{ℤ_{2}} are AF algebras.

This result generalizes Theorem 3.1 of [6] and completely answers a question raised in the introduction of [28].

In Section 1, we give the necessary background on twisted group algebras and the proof of Theorem 0.3. The realization of the crossed products A_{θ} ⋊_{α} F as twisted group algebras of the group ℤ^{2} ⋊ F is done in Section 2. The K-theory computations for C_{r}^{∗}(ℤ^{2} ⋊ F) are done in Sections 3 and 4. Section 5 contains the proof that the relevant actions have the tracial Rokhlin property. In Section 6, we prove the Universal Coefficient Theorem and put everything together. We also prove Theorem 0.4.

This paper contains the main result of Section 10 of the unpublished long preprint [58], Theorem 0.4 here. Although the three steps, as described at the beginning of the introduction, are the same, the proofs of all three of them differ substantially from the proofs given in [58]. This paper also supersedes Sections 8 and 9 of [58]. Again, the proofs of Steps (2) and (3) differ substantially from those in [58]. We improve on [58] by calculating the K-theory of the crossed products A_{θ} ⋊_{α} F. No K-theory calculations were done in [58], so that the crossed products were merely shown to be simple AH algebras with real rank zero for F = ℤ_{3}, ℤ_{4}, ℤ_{6} and all irrational θ, and to be AF algebras for F = ℤ_{4} and a dense G_{δ}-set of values of θ. (The case F = ℤ_{2} was already known, and in any case it is covered by Theorem 0.4.)

The third author would like to thank Masaki Izumi for useful discussions concerning the tracial Rokhlin property.

1. Twisted group algebras and actions on K

We begin by recalling some basic facts about group C*-algebras twisted by cocycles. This is a special case of the theory of crossed product C*-algebras twisted by cocycles of [52]. Also see [77] for the case in which the group is discrete.

Let G be a second countable locally compact group, with modular function Δ_{G}: G → (0, ∞). Let ω: G×G → 𝕋 be a Borel 2-cocycle on G. (Recall the algebraic conditions: ω(s, t)ω(r, st) = ω(r, s)ω(rs, t) and ω(1, s) = ω(s, 1) = 1 for r, s, t ∈ G.) Then the twisted convolution algebra L^{1}(G, ω) is defined to be the vector space of all integrable complex functions on G with convolution and involution given by

(f ∗_{ω} g)(s) = ∫_{G} f(t)g(t^{−1}s)ω(t, t^{−1}s) dt and f^{∗}(s) = Δ_{G}(s^{−1})\overline{ω(s, s^{−1})f(s^{−1})}.

An ω-representation of G on a Hilbert space H is a Borel map V : G → U(H), the unitary group of H, with respect to the strong operator topology on U(H), satisfying

V (s)V (t) = ω(s, t)V (st)

far all s, t ∈ G. The regular ω-representation of G is the representation L_{ω}: G → U(L^{2}(G)) given by the formula

(L_{ω}(s)ξ)(t) = ω(s, s^{−1}t)ξ(s^{−1}t)

for ξ ∈ L^{2}(G). Every ω-representation V : G → U(H) determines a contractive ∗-homomorphism, also denoted V, from L^{1}(G, ω) to B(H) via the formula

V (f) = ∫_{G} f(s)V (s) ds

for f ∈ L^{1}(G, ω), and every nondegenerate representation of L^{1}(G, ω) appears in this way. The full twisted group algebra C^{∗}(G, ω) is defined to be the enveloping C*-algebra of L^{1}(G, ω) and the reduced twisted group algebra C_{r}^{∗}(G, ω) is defined to be the image of C^{∗}(G, ω) under the regular ω-representation L_{ω}. If G is amenable, then both algebras coincide (Theorem 3.11 of [52]). In this case we simply write C^{∗}(G, ω).

Recall that two cocycles ω, ω′ ∈ Z^{2}(G, 𝕋) are cohomologous if there exists a Borel function u: G → 𝕋 such that, with ∂u(s, t) = u(s)u(t)\overline{u(st)}, we have ω′ = ∂u · ω. (See Chapter 1 of [48]; the action of G on 𝕋 is trivial.) It is then easy to verify that the map L^{1}(G, ω′) → L^{1}(G, ω), given by f ↦ u · f, extends to isomorphisms of the full and reduced twisted group C*-algebras.

In this section, let K denote the C*-algebra of compact operators on a separable infinite dimensional Hilbert space H, let U be the group of unitary operators on H, and let PU = U/𝕋1 denote the projective unitary group. If V : G → U is an ω-representation, then V determines an action α: G → PU ≅ Aut(K) by α_{s} = Ad(V (s)) for s ∈ G. (Continuity follows from Proposition 5(a) of [50].) With \overline{ω}(s, t) = \overline{ω(s, t)}, the full and reduced crossed products K ⋊_{α} G and K ⋊_{α,r} G are then isomorphic to C^{∗}(G, \overline{ω}) ⊗ K and C_{r}^{∗}(G, \overline{ω}) ⊗ K. On the level of L^{1}-algebras the isomorphisms are given by the map Φ: L^{1}(G, \overline{ω}) ⊙ K → L^{1}(G, K) determined by

(1.1)  Φ(f ⊗ k)(s) = f(s)kV (s)^{∗}

for f ∈ L^{1}(G, \overline{ω}), k ∈ K, and s ∈ G. (See for example Theorem 1.4.15 and Example 1.1.4 of [19], but this is easily proved directly.) Conversely, if α: G → Aut(K) is any action of G on K, then by choosing a Borel section c: PU → U we obtain a Borel map V^{α} = c ∘ α: G → U such that α_{s} = Ad(V^{α}(s)) for all s ∈ G. It is then easy to check that there exists a Borel cocycle ω_{α} ∈ Z^{2}(G, 𝕋) such that V^{α}(s)V^{α}(t) = ω_{α}(s, t)V^{α}(st) for all s, t ∈ G. Thus, any action α: G → Aut(K) is given by an ω_{α}-representation for some cocycle ω_{α} ∈ Z^{2}(G, 𝕋).

If A is any C*-algebra, then two actions α, α′: G → Aut(A) are called exterior equivalent if there exists a strictly continuous map v: G → UM(A) (the unitary group in the multiplier algebra of A) such that α′_{s} = Ad(v_{s}) ∘ α_{s} and v_{st} = v_{s}α_{s}(v_{t}) for all s, t ∈ G. (Compare with 8.11.3 of [54].) It is easily seen that exterior equivalent actions have isomorphic crossed products and isomorphic reduced crossed products, with isomorphisms given on L^{1}(G, A) by f ↦ f · v.

Following [21], if X is any locally compact space we denote by E_{G}(X) the set of all exterior equivalence classes of C_{0}(X)-linear actions of G on C_{0}(X, K), that is, actions α: G → Aut (C_{0}(X, K)) such that α_{s}(f · g) = f · α_{s}(g) for all f ∈ C_{0}(X) and g ∈ C_{0}(X, K). Any such action is completely determined by its evaluations α^{x}: G → Aut(K) for x ∈ X, defined by α_{s}^{x}(g(x)) = (α_{s}(g))(x) for g ∈ C_{0}(X, K) and x ∈ X. Thus, we should understand a C_{0}(X)-linear action as a continuous family of actions on K. It is shown in [17] (see Lemma 3.1 and Theorem 3.6) and [21] that E_{G}(X) is a group with multiplication [α] · [β] = [α ⊗_{X} β], where (α ⊗_{X} β)^{x} = α^{x} ⊗ β^{x} for x ∈ X. The following result, basically due to Mackey, is well known; for example see 6.3 in [17]:

Proposition 1.1. Let G be a second countable locally compact group. Two actions α and α′ of G on K are exterior equivalent if and only if [ω_{α}] = [ω_{α′}] ∈ H^{2}(G, 𝕋). Moreover, the mapping [α] ↦ [ω_{α}] is a bijective homomorphism from E_{G}(pt) to H^{2}(G, 𝕋).

Proof. A calculation, which we omit, shows that if α and α′ are two actions with corresponding cocycles ω and ω′, then α is exterior equivalent to α′ if and only if ω is cohomologous to ω′. This shows that [α] ↦ [ω_{α}] is well defined and injective. For surjectivity, given ω, take α to be conjugation by the regular ω-representation of G on L^{2}(G) (or L^{2}(G)⊗H if G is finite). (Continuity of α follows from Proposition 5(a) of [50].) □

We write K_{ω} if we consider K equipped with an action corresponding to ω as above.

For an action α: G → Aut(A), we denote by K_{∗}^{top}(G; A) the left hand side of the Baum-Connes Conjecture with coefficients A. See Section 9 of [3], where it is called K_{∗}^{G}(\underline{E}G; A). As there, we let μ: K_{∗}^{top}(G; A) → K_{∗}(A⋊_{α,r} G) be the assembly map. Versions of the following definition have appeared in [45, 11].

Definition 1.2. Let [ω] ∈ H^{2}(G, 𝕋). Then the twisted topological K-theory of G with respect to [ω] is defined to be the topological K-theory K_{∗}^{top}(G; ω) = K_{∗}^{top}(G; K\overline{ω}). The twisted assembly map for G with respect to ω is then defined to be

μ_{ω}: K_{∗}^{top}(G; K\overline{ω}) → K_{∗}(K\overline{ω} ⋊_{r} G) ⟶^{≅} K_{∗}(C_{r}^{∗}(G, ω)).

Up to obvious isomorphisms, the definition does not depend on the choice of the representative ω of the class [ω] ∈ H^{2}(G, 𝕋) and the representative α of the class in E_{G}(pt) corresponding to \overline{ω}.

Of course, if G is a group which satisfies the Baum-Connes conjecture with coefficients, then the twisted assembly map is an isomorphism for all ω ∈ Z^{2}(G, 𝕋). But it is sometimes possible to show that the Baum-Connes conjecture holds for K (with respect to any action of G on K) without knowing that the conjecture holds for all coefficients. For example, it is shown in [14] that every almost connected group satisfies the conjecture for K but the Baum-Connes conjecture with arbitrary coefficients is not known in general for those groups.

We now consider homotopies between cocycles. The definition of the twisted assembly map suggests that it is actually useful to consider homotopies for actions on K. The first part of the following definition is a special case, in different language, of Definition 3.1 of [57].

Definition 1.3. A homotopy of actions on K is a C([0, 1])-linear action of G on C([0, 1], K). A homotopy between two classes [β^{0}], [β^{1}] ∈ E_{G}(pt) is a class [β] ∈ E_{G}([0, 1]) with evaluations [β^{0}] and [β^{1}] at the points 0 and 1.

Remark 1.4. The notion of homotopy in Definition 1.3 is, at least in principle, weaker than the notion of homotopy between classes in H^{2}(G, 𝕋) ≅ E_{G}(pt) used in Theorem 0.3. Indeed, let Ω: G × G → C([0, 1], 𝕋) be a 2-cocycle with evaluations ω_{x} = Ω( · , ·)(x) for x ∈ [0, 1]. By Proposition 3.1 of [34], and with L_{ω_{x}} being the regular ω_{x}-representation on K(L^{2}(G)), the formula

α_{s}(g)(x) = L_{ω_{x}}(s)g(x)L_{ω_{x}}(s)^{∗},

for g ∈ C([0, 1], K), defines a continuous action α: G → Aut (C([0, 1], K)). Thus, if [α_{0}], [α_{1}] ∈ E_{G}(pt) correspond to [ω_{0}], [ω_{1}] ∈ H^{2}(G, 𝕋) under the correspondence described in Proposition 1.1, we see that [α_{0}] and [α_{1}] are homotopic classes in E_{G}(pt).

Using the results of [21, 20] one can show that, conversely, homotopy of classes in E_{G}(pt) implies homotopy of the corresponding cocycles for a very large class of groups. This class includes all almost connected groups by Theorem 1.4 of [20] and all smooth groups in the sense of [49] by Theorem 5.4 of [21]; note that it is shown in [49] that all discrete groups and all compact groups are smooth. However, it is not clear to us whether this direction holds in general.

An even weaker notion of homotopy of classes in E_{G}(pt) ≅ H^{2}(G, 𝕋) is obtained by using the topology on H^{2}(G, 𝕋) defined by Moore in [50] and defining two classes to be homotopic if they can be connected via a continuous path γ: [0, 1] → H^{2}(G, 𝕋). Again, Theorem 1.4 of [20] and Theorem 5.4 of [21] imply that this notion of homotopy coincides with the previous notions whenever G is smooth or almost connected, but we do not know this in general. The problem is that the topology of H^{2}(G, 𝕋), which is defined via pointwise almost everywhere convergence of cocycles, can have rather poor separation properties.

The structure of the group E_{G}(X) of all exterior equivalence classes of C_{0}(X)- linear actions on C_{0}(X, K) was extensively studied in [21, 20]. As an easy corollary of the results obtained in [21] we get:

Proposition 1.5. Suppose that G is compact and X is contractible. Then, for any x ∈ X, the evaluation map E_{G}(X) → E_{G}(pt), given by [α] ↦ [α^{x}], is an isomorphism. In particular, every C_{0}(X)-linear action α: G → Aut (C_{0}(X, K)) is exterior equivalent to the diagonal action id_{X} ⊗α^{x} on C_{0}(X) ⊗ K ≅ C_{0}(X, K).

Proof. Since compact groups are smooth in the sense of Moore (see the discussion preceding Proposition 3.1 in [49] and also Remark 4.2 of [21]), it follows from Theorem 5.4 of [21] and its proof that there is a short exact sequence

1 → Ȟ^{1}(X; Ĝ_{ab}) → E_{G}(X) → C(X, H^{2}(G, 𝕋)) → 1,

where Ȟ^{1}(X; Ĝ_{ab}) denotes Čech cohomology with coefficients in the (discrete) dual group of the abelianization G_{ab} of G and H^{2}(G, 𝕋) is topologized as in [50]. The quotient map in this sequence sends a class [α] ∈ E_{G}(X) to the mapping x ↦ [α^{x}] ∈ E_{G}(pt) ≅ H^{2}(G, 𝕋). Since X is contractible, Ȟ^{1}(X; Ĝ_{ab}) = 0. Since G is compact, H^{2}(G, 𝕋) is countable by Corollary 1, on page 56, of [48]. Since it is locally compact Hausdorff (see Remark 4.2 of [21]), H^{2}(G, 𝕋) is discrete. Since X is connected, it follows that C(X, H^{2}(G, 𝕋)) ≅ H^{2}(G, 𝕋). This completes the proof. □

The following result follows from Theorem 1.5 of [15].

Proposition 1.6. Let G be a second countable locally compact group and let A and B be G-C*-algebras. Let z ∈ KK_{0}^{G}(A, B) have the property that for all compact subgroups L of G, the Kasparov product with the restriction res_{L}^{G}(z) ∈ KK_{0}^{L}(A, B) induces bijective homomorphisms KK_{∗}^{L}(ℂ, A) → KK_{∗}^{L}(ℂ, B). Then the Kasparov product with z induces an isomorphism K_{∗}^{top}(G; A) ≅ K_{∗}^{top}(G; B).

Proof. For a closed subgroup H of G and a proper H-space Y we define F_{H}^{n}(C_{0}(Y)) = KK_{n}^{G}(C_{0}(Y), A) and G_{H}^{n}(C_{0}(Y)) = KK_{n}^{G}(C_{0}(Y), B). Let S(G) be the collection of subgroups of G defined in the introduction to Section 1 of [15]. Then, by Remark 1.2 of [15], the functors F^{∗} = (F_{H}^{∗})_{H∈S(G)} and G^{∗} = (G_{H}^{∗})_{H∈S(G)} are going-down functors in the sense of Definition 1.1 of [15]. For any z ∈ KK_{0}^{G}(A, B), the family of transformations

F_{H}^{n}(C_{0}(Y)) = KK_{n}^{H}(C_{0}(Y), A) ⟶^{⊗_{A} res_{H}^{G}(z)} KK_{n}^{H}(C_{0}(Y), B) = G_{H}^{n}(C_{0}(Y))

is a going-down transformation in the sense of Definition 1.4 of [15]. Indeed, the assumptions (1) and (2) of that definition follow from naturality of the Kasparov product. Thus, the proof of the proposition will follow from Theorem 1.5 of [15] if we know that

(1.2)  KK_{n}^{L}(C_{0}(V), A) ⟶^{⊗_{A} res_{L}^{G}(z)} KK_{n}^{L}(C_{0}(V), B)

is an isomorphism for all compact subgroups L of G and for all real finite dimensional representation spaces V of L.

Let V be a real finite dimensional representation space of L. Lemma 7.7(2) and Lemma 7.7(1) of [12], in order, give isomorphisms

KK_{n}^{L}(C_{0}(V), D) ⟶^{≅} KK_{n}^{L}(C_{0}(V) ⊗ C_{0}(V), C_{0}(V) ⊗ D) ⟶^{≅} KK_{n}^{L}(ℂ, C_{0}(V) ⊗ D)

for any D, which respect Kasparov products. To prove that (1.2) is an isomorphism, it is therefore enough to show that Kasparov product with 1_{C_{0}(V)}⊗_{ℂ}res_{L}^{G}(z) induces an isomorphism

(1.3)  KK_{n}^{L}(ℂ, C_{0}(V) ⊗ A) ≅ KK_{n}^{L}(ℂ, C_{0}(V) ⊗ B).

We prove this by a bootstrap process.

First, let L′ ⊆ L ⊆ G be compact subgroups of G. We claim that multiplication with 1_{C(L/L′)} ⊗_{ℂ} res_{L}^{G}(z) induces isomorphisms KK_{n}^{L}(ℂ, C(L/L′) ⊗ A) ≅ KK_{n}^{L}(ℂ, C(L/L′) ⊗ B). For any G-C*-algebra D, let q_{D}: C(L/L′) ⊗ D → D be the quotient map given by evaluation at the identity coset L′ ∈ L/L′. By Theorem 20.5.5 of [5], the chain of maps

KK_{n}^{L}(ℂ, C(L/L′) ⊗ D) ⟶^{res_{L′}^{L}} KK_{n}^{L′}(ℂ, C(L/L′) ⊗ D) ⟶^{q_{∗}} KK_{n}^{L′}(ℂ, D)

is an isomorphism. Moreover, when applied to A and B in place of D, it transforms Kasparov product with 1_{C(L/L′)} ⊗_{ℂ} res_{L}^{G}(z) to Kasparov product with res_{L′}^{G}(z). The claim thus follows from our assumption on z.

Now let V be any real finite dimensional representation space of L. Let

∅ = U_{0} ⊆ U_{1} ⊆ ⋯ ⊆ U_{l} = V

be a stratification as in the proof of Lemma 2.8 of [15]. Thus, the U_{j} are L-invariant open subsets of V such that the difference sets M_{j} = U_{j} ∖ U_{j−1} and their quotients M_{j}/L are smooth manifolds, and M_{j} is locally L-homeomorphic to spaces of the form X × L/L_{j} for suitable compact subgroups L_{j} ⊆ L. Now M_{j}/L is triangulable by Theorem 10.6 of [46], and refining the triangulation allows us to assume that the inverse image of each open simplex S is L-homeomorphic to S × L/L_{j} with the trivial action of L on S. By Bott periodicity and the previous paragraph, (1.3) is an isomorphism when C_{0}(V) is replaced by C_{0}(S × L/L′). Using six term exact sequences and the Five Lemma, we conclude that (1.3) is an isomorphism when C_{0}(V) is replaced by C_{0}(X) for the inverse image X in M_{j} of any finite complex in the triangulation of M_{j}/L, or the inverse image of the topological interior of one. Using continuity of equivariant K-theory for compact group actions (Theorem 2.8.3(6) of [56] or Proposition 11.9.2 of [5]), we find that (1.3) is an isomorphism when C_{0}(V) is replaced by C_{0}(M_{j}). Again using six term exact sequences and the Five Lemma, we conclude that (1.3) is an isomorphism for our given real representation space V. □

Theorem 1.7. Let α: G → Aut (C([0, 1], K)) be a homotopy of actions on K. Write K_{x} for K when we consider K as the fiber at x equipped with the action α^{x}: G → Aut(K), and let q_{x}: C([0, 1], K) → K_{x} denote evaluation at x. Then

(q_{x})_{∗}: K_{∗}^{top}(G; C([0, 1], K)) → K_{∗}^{top}(G; K_{x})

is an isomorphism for all x ∈ [0, 1].

Proof. By Proposition 1.6 it is enough to check that

(q_{x})_{∗}: KK_{∗}^{L}(ℂ, C([0, 1], K)) → KK_{∗}^{L}(ℂ, K_{x})

is an isomorphism for every compact subgroup L ⊆ G. By the Green-Julg theorem (Theorem 11.7.1 of [5] or Theorem 2.8.3(7) of [56]), this is equivalent to saying that

(q_{x} ⋊ L)_{∗}: K_{∗}(C([0, 1], K) ⋊_{α} L) → K_{∗}(K_{x} ⋊_{α^{x}} L)

is an isomorphism for all such L. We know from Proposition 1.5 that the restriction res_{L}^{G}(α) of α to L is exterior equivalent to id_{[0,1]} ⊗ res_{L}^{G}(α^{x}). So C([0, 1], K) ⋊_{α} L ≅ C([0, 1], K_{x}⋊_{α^{x}} L), and it is easy to check that q_{x}⋊L corresponds to the evaluation map ev_{x}: C([0, 1], K_{x} ⋊_{α^{x}} L) → K_{x} ⋊_{α^{x}} L. This map induces an isomorphism on K-theory because it is a homotopy equivalence. □

Corollary 1.8. Let α: G → Aut (C([0, 1], K)) be a homotopy for actions on K and assume that G satisfies the Baum-Connes conjecture with coefficients C([0, 1], K) (with respect to α) and with coefficients K_{x} for all x ∈ [0, 1]. Then, for any x ∈ [0, 1], the map

(q_{x} ⋊ G)_{∗}: K_{∗}(C([0, 1], K) ⋊_{α,r} G) → K_{∗}(K_{x} ⋊_{α^{x},r} G)

is an isomorphism.

Corollary 1.8 fails if K is replaced by a general AF algebra, or by a general commutative C*-algebra, even for G = ℤ_{2}. See Examples 3.3 and 3.5 of [57]. Sometimes something can be said; see Theorem 4.3 of [57].

Theorem 1.9 (Theorem 0.3). Suppose that ω_{0}, ω_{1} ∈ Z^{2}(G, 𝕋) are homotopic Borel 2-cocycles on the second countable locally compact group G, that is, there exists a Borel 2-cocycle Ω ∈ Z^{2}(G, C([0, 1], 𝕋)) such that ω_{j} = Ω( · , ·)(j) for j = 0, 1. Suppose further that G satisfies the Baum-Connes conjecture with coefficients. Then K_{∗}(C_{r}^{∗}(G, ω_{0})) ≅ K_{∗}(C_{r}^{∗}(G, ω_{1})).

Proof. If G satisfies the Baum-Connes conjecture with arbitrary coefficients, then G certainly satisfies the hypotheses of Corollary 1.8. The proof follows by combining this corollary with the isomorphism (1.1) and Remark 1.4. □

Remark 1.10. Let us briefly discuss the assumption on the Baum-Connes conjecture as given in Corollary 1.8. If G satisfies the conjecture with coefficients C([0, 1], K) for all homotopies α: G → Aut (C([0, 1], K)) (we then say that G satisfies the conjecture for homotopies), then G automatically satisfies the conjecture with coefficients K for any G-action on K. This follows because for any action β of G on K, the action id_{[0,1]} ⊗β of G on C([0, 1], K) is equivariantly KK-equivalent to K_{β}. Conversely, if we just know that G satisfies the conjecture for K with respect to any given G-action, then it follows from Proposition 3.1 of [14] that G satisfies the conjecture for homotopies as soon as G is exact and has a γ-element in the sense of Kasparov.

For later use it is convenient to restate Corollary 1.8 completely in terms of homotopies between cocycles. We will need the reduced twisted crossed product C*-algebra C([0, 1])⋊_{Ω,r} G for a cocycle Ω ∈ Z^{2}(G, C([0, 1], 𝕋)). It is a completion of the convolution algebra L^{1}(G, C([0, 1]), Ω) of all C([0, 1])-valued L^{1}-functions on G with convolution given by

f ∗_{Ω} g(s, x) = ∫_{G} f(t, x)g(t^{−1}s, x)Ω(t, t^{−1}s)(x) dx.

See [52], and, for the case of a discrete group, [77].

Corollary 1.11. Suppose that G satisfies the Baum-Connes conjecture with respect to any homotopy of actions on K. Let G act trivially on C([0, 1], 𝕋). Then for any Ω ∈ Z^{2}(G, C([0, 1], 𝕋)) and any x ∈ [0, 1], the canonical quotient map p_{x}: C([0, 1]) ⋊_{Ω,r} G → C_{r}^{∗}(G, ω_{x}) induces an isomorphism on K-theory.

Proof. As explained in Remark 1.4, the cocycle Ω determines a homotopy α: G → Aut (C([0, 1], K)). We claim that, if \overline{α}: G → Aut (C([0, 1], K)) is the homotopy corresponding to the inverse \overline{Ω} of Ω, then there is an isomorphism

(C([0, 1]) ⋊_{Ω,r} G) ⊗ K ≅ C([0, 1], K) ⋊_{\overline{α},r} G.

For the full twisted crossed products, this follows from Proposition 4.6 of [22] (which is a special case of Theorem 3.4 of [52]). Remark 3.12 and the proof of Theorem 3.11 of [52] imply that this is correct for the reduced crossed products as well.

With ω_{x} = Ω( · , ·)(x) for x ∈ [0, 1], there are obvious quotient maps p_{x}: C([0, 1]) ⋊_{Ω,r} G → C_{r}^{∗}(G, ω_{x}). Moreover, with the right vertical arrow being the isomorphism of (1.1), the diagram

(C([0, 1]) ⋊_{Ω,r} G) ⊗ K ⟶^{p_{x}⊗Id_{K}} C_{r}^{∗}(G, ω_{x}) ⊗ K
≅↓  ↓≅
C([0, 1], K) ⋊_{\overline{α},r} G ⟶_{q_{x}⋊G} K ⋊_{\overline{α}^{x},r} G

commutes. Using the induced maps in K-theory for this diagram together with Corollary 1.8 and Remark 1.10, the result follows. □

Definition 1.12. We call a class [ω] ∈ H^{2}(G, 𝕋) real if there exists a cocycle c ∈ Z^{2}(G, ℝ) such that ω is equivalent to the cocycle (s, t) ↦ exp (ic(s, t)).

In this case we obtain a homotopy Ω ∈ Z^{2}(G, C([0, 1], 𝕋)) between [ω] and the trivial cocycle [1] by the formula

Ω(s, t)(x) = e^{ix·c(s,t)}

for x ∈ [0, 1] and s, t ∈ G. As a consequence we get the following corollary. Several special cases of it appear in [24, 53, 11, 43, 44]. (In [45], Mathai states a theorem saying that for any discrete G and any real 2-cocycle ω on G, one has K_{0}(C_{r}^{∗}(G, ω)) ≅ K_{0}(C_{r}^{∗}(G)). Unfortunately, the proof given in [45] has a substantial gap, and hence the result is not available so far.)

Corollary 1.13. Let G be a group which satisfies the Baum-Connes conjecture for homotopies of actions on K, and let ω ∈ Z^{2}(G, 𝕋) be a cocycle such that [ω] is real. Then K_{∗}(C_{r}^{∗}(G, ω)) ≅ K_{∗}(C_{r}^{∗}(G)). More generally, if G is any second countable locally compact group and [ω] ∈ H^{2}(G, 𝕋) is real, then K_{∗}^{top}(G, ω) ≅ K_{∗}^{top}(G).

Let Θ be any skew symmetric real d × d matrix. Then Θ determines a 2-cocycle on ℤ^{d} via

ω_{Θ}(m, n) = exp(πi⟨Θm, n⟩)

for m, n ∈ ℤ^{d}. The corresponding d-dimensional noncommutative torus [65] is the twisted group C*-algebra A_{Θ} = C^{∗}(ℤ^{d}, ω_{Θ}). Equivalently, A_{Θ} is the universal C*- algebra generated by unitaries u_{1}, u_{2}, …, u_{d} subject to the relations

u_{k}u_{j} = exp(2πiθ_{j,k})u_{j}u_{k}

for 1 ≤ j, k ≤ d. (Of course, if all θ_{j,k} are integers, it is not really noncommutative.)

Since ω_{Θ} is a real cocycle, Corollary 1.13 immediately gives the following well known computation of the K-theory of a higher dimensional noncommutative torus, as found, for example, in [24] and [64]. Of course, we do not immediately get the additional information given in [24] and [64].

Corollary 1.14. For any d ≥ 1 and any skew symmetric real d × d matrix Θ, we have K_{j}(C^{∗}(ℤ^{d}, ω_{Θ})) ≅ ℤ^{2^{d−1}} for j = 0, 1.

It is shown in Theorem 3.2 of [1] (see Section 2 of [2] for terminology) that every class in H^{2}(ℤ^{d}, 𝕋) has a representative of the form ω_{Θ}(n, m) = e^{i⟨Θn,m⟩} with Θ a skew symmetric real d × d matrix.

2. Twisted group C*-algebras of semidirect products

As mentioned in the introduction, the rotation algebra A_{θ} can be realized as the twisted group algebra C^{∗}(ℤ^{2}, ω_{θ}) with

ω_{θ} (( \begin{matrix}n \\ m \end{matrix}) , ( \begin{matrix}n′ \\ m′ \end{matrix} )) = e^{πiθ(n′m−nm′)}.

Indeed, if we define u_{θ} = δ( \begin{matrix}1 \\ 0 \end{matrix} ) and v_{θ} = δ( \begin{matrix}0 \\ 1 \end{matrix} ), then u_{θ} and v_{θ} are unitary generators

of C^{∗}(ℤ^{2}, ω_{θ}) which satisfy the commutation relation v_{θ}u_{θ} = e^{2πiθ}u_{θ}v_{θ}. This (and other computations below) follows because ( \begin{matrix}n \\ m \end{matrix}) ↦ δ( \begin{matrix}n \\ m \end{matrix}) ∈ C^{∗}(ℤ^{2}, ω_{θ}) is an ω_{θ}-

representation of ℤ^{2}.

As explained in the introduction, we want to study the crossed products A_{θ} ⋊_{α} F with F a finite subgroup of SL_{2}(ℤ) and the action α^{θ} of SL_{2}(ℤ) on on A_{θ} as given in (0.1). We will realize the crossed product A_{θ} ⋊_{α^{θ}} H, for any subgroup H ⊆ SL_{2}(ℤ), as a twisted group algebra C^{∗}(ℤ^{2} ⋊ H, ω̃_{θ}), where H ⊆ SL_{2}(ℤ) acts on ℤ^{2} via matrix multiplication and ω̃_{θ} is a suitable extension of ω_{θ} to ℤ^{2} ⋊ H. A short computation shows that the cocycle ω_{θ} ∈ Z^{2}(ℤ^{2}, 𝕋) is invariant under the action of SL_{2}(ℤ) on ℤ^{2}, that is

ω_{θ} (N ( \begin{matrix}n \\ m \end{matrix}) , N ( \begin{matrix}n′ \\ m′ \end{matrix} )) = ω_{θ} (( \begin{matrix}n \\ m \end{matrix}) , ( \begin{matrix}n′ \\ m′ \end{matrix} ))

for all N ∈ SL_{2}(ℤ). This implies that the action of SL_{2}(ℤ) on ℤ^{2} defines an action β^{θ}: SL_{2}(ℤ) → Aut (C^{∗}(ℤ^{2}, ω_{θ})) via

β_{N}^{θ}(f) ( \begin{matrix}n \\ m \end{matrix}) = f (N^{−1} ( \begin{matrix}n \\ m \end{matrix}))

for f in the dense subalgebra l^{1}(ℤ^{2}, ω_{θ}). It follows from this formula and the fact that ( \begin{matrix}n \\ m \end{matrix}) ↦ δ( \begin{matrix}n \\ m \end{matrix}) ∈ C^{∗}(ℤ^{2}, ω_{θ}) is an ω_{θ}-representation that

β_{N}^{θ}(u_{θ}) = β_{N}^{θ}(δ( \begin{matrix}1 \\ 0 \end{matrix} )) = δ( \begin{matrix}n_{1,1} \\ n_{2,1} \end{matrix} ) = \overline{ω_{θ} (( \begin{matrix}n_{1,1} \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ n_{21} \end{matrix} ))}δ( \begin{matrix}n_{1,1} \\ 0 \end{matrix} )δ_{( \begin{matrix}0 \\ n_{2,1} \end{matrix} )}

= e^{πin_{11}n_{21}θ}(δ( \begin{matrix}1 \\ 0 \end{matrix} ))^{n_{1,1}}(δ( \begin{matrix}0 \\ 1 \end{matrix} ))^{n_{2,2}} = e^{πin_{1,1}n_{2,1}θ}u_{θ}^{n_{1,1}}v_{θ}^{n_{2,1}}

and, similarly, β_{N}^{θ}(v_{θ}) = e^{πin_{1,2}n_{2,2}θ}u_{θ}^{n_{1,2}}v_{θ}^{n_{2,2}}, so β^{θ} = α^{θ} for all θ ∈ [0, 1].

We recall some facts about semidirect products. Let M and H be locally compact groups; in view of our intended application (M = ℤ^{2} and H ⊆ SL_{2}(ℤ)), we write the group operation on M additively and the operation on H multiplicatively. Assume further that H acts continuously on M by automorphisms; we write the action as (h, m) ↦ hm for h ∈ H and m ∈ M. Then the semidirect product M ⋊ H is the set M × H, equipped with the product topology and the multiplication (m, h) · (m′, h′) = (m + hm′, hh′). We let μ: H → (0, ∞) be the function such that ∫_{M} f(m) dm = μ(h) ∫_{M} f(h^{−1} · m) dm for all f ∈ L^{1}(M), where dm and dh denote left Haar measures on M and H. Recall that left Haar measure on M ⋊ H is given by the formula

∫_{M⋊H} f(m, h) d(m, h) = ∫_{H} ∫_{M} f(m, h)μ(h) dm dh.

We will also need a continuously parametrized version of the following lemma. A general statement is awkward to formulate, but what we actually need (see Remark 2.3 below) has essentially the same proof, so we just refer to the proof.

Lemma 2.1. Suppose that M ⋊ H is a semidirect product of the second countable locally compact groups M and H and assume that ω ∈ Z^{2}(M, 𝕋) is invariant under the action of H on M, that is, ω(h · n, h · m) = ω(n, m) for all n, m ∈ M, h ∈ H. Then:

(1) There is a cocycle ω̃ ∈ Z^{2}(M ⋊ H, 𝕋) defined by

ω̃((m, h), (m′, h′)) = ω(m, h · m′).

(2) There is an action α: H → Aut(C^{∗}(M, ω)) (and similarly for C_{r}^{∗}(M, ω)) determined by α_{h}(f)(m) = μ(h)f(h^{−1}m) for f ∈ L^{1}(M, ω).

(3) There are isomorphisms

C^{∗}(M ⋊ H, ω̃) ≅ C^{∗}(M, ω) ⋊_{α} H and C_{r}^{∗}(M ⋊ H, ω̃) ≅ C_{r}^{∗}(M, ω) ⋊_{α,r} H

given on the level of L^{1}-functions by f ↦ Φ(f) ∈ L^{1}(H, L^{1}(M, ω)) with

Φ(f)(h) = μ(h)f( · , h).

Proof. Part (1) follows from straightforward computations. For full twisted group algebras, Parts (2) and (3) are very special cases of Theorem 4.1 of [52]. Since the deduction from that result is quite tedious, we sketch the proof of (2) and (3) in the case of reduced twisted group algebras. Note first that we have a faithful representation L_{ω}^{M}: C_{r}^{∗}(M, ω) → B(L^{2}(M)) given by L_{ω}^{M}(f)ξ = f ∗_{ω} ξ for f ∈ L^{1}(M, ω) and ξ ∈ L^{2}(M). It is easy to check on L^{1}-functions that α_{h}: L^{1}(M, ω) → L^{1}(M, ω) preserves all algebraic operations. Define a unitary operator U_{h}: L^{2}(M) → L^{2}(M) by (U_{h}ξ)(m) = \sqrt{μ(h)}ξ(h^{−1} · m). We check, as a sample, that U_{h}^{∗}L_{ω}^{M}(α_{h}(f))U_{h} = L_{ω}^{M}(f) for all f ∈ L^{1}(M, ω), which then implies that α_{h} is norm preserving. Let ξ ∈ L^{2}(M). Then for almost all n ∈ M, we have, using H-invariance of ω at the fourth step and the definition of μ at the fifth step,

(U_{h}^{∗}L_{ω}^{M}(α_{h}(f))U_{h}ξ)(n) = \sqrt{μ(h^{−1})}(α_{h}(f) ∗_{ω} U_{h}(ξ))(h · n)
= \sqrt{μ(h^{−1})} ∫_{M} α_{h}(f)(m)U_{h}(ξ)(h · n − m)ω(m, h · n − m) dm
= μ(h) ∫_{M} f(h^{−1} · m)ξ(n − h^{−1} · m)ω(m, h · n − m) dm
= μ(h) ∫_{M} f(h^{−1} · m)ξ(n − h^{−1} · m)ω(h^{−1} · m, n − h^{−1} · m) dm
= ∫_{M} f(m)ξ(n − m)ω(m, n − m) dm = (L_{ω}^{M}(f)ξ)(n).

This proves (2). To check the isomorphism of (3) in the reduced case one first checks, using Fubini’s theorem, that the map Φ: L^{1}(M ⋊ H, ω̃) → L^{1}(H, L^{1}(M, ω)) is a bijection which preserves all algebraic operations. To see that it extends to an isomorphism of the reduced C*-algebras, consider the unitary V : L^{2}(M ⋊ H) → L^{2}(H, L^{2}(M)) given by (V ξ)(h)(n) = ξ(h · n, h). Let LM⋊H ; ω̃ denote the ω̃-regular representation of M ⋊ H. Let Λ: L^{1}(H, C_{r}^{∗}(M, ω)) → B(L^{2}(H, L^{2}(M))) denote the integrated from of the regular representation of (C_{r}^{∗}(M, ω), H, α) induced from the faithful representation L_{ω}^{M}: C_{r}^{∗}(M, ω) → B(L^{2}(M)). It is given by

(Λ(g)η)(h) = ∫_{H} L_{ω}^{M}(α_{h^{−1}}(g(l))η(l^{−1}h) dl

for g ∈ L^{1}(H, C_{r}^{∗}(M, ω)) and η ∈ L^{2}(H, L^{2}(M)). Using this formula, one can check that

V LM⋊H ; ω̃(f)ξ = Λ(Φ(f))V ξ

for all f ∈ L^{1}(M ⋊ H, ω̃) and ξ ∈ L^{2}(M ⋊ H). This implies that Φ preserves the C*-norm. □

Corollary 2.2. Suppose that H is a subgroup of SL_{2}(ℤ). Define a cocycle on ℤ^{2}⋊H by

ω̃_{θ} (( ( \begin{matrix}n \\ m \end{matrix}) , N), ( ( \begin{matrix}n′ \\ m′ \end{matrix} ) , N′)) = ω_{θ} (( \begin{matrix}n \\ m \end{matrix}) , N · ( \begin{matrix}n′ \\ m′ \end{matrix} ))

for ( \begin{matrix}n \\ m \end{matrix}) , ( \begin{matrix}n′ \\ m′ \end{matrix} ) ∈ ℤ^{2} and N, N′ ∈ H. Then there is a canonical isomorphism

A_{θ} ⋊_{α^{θ},r} H = C^{∗}(ℤ^{2}, ω_{θ}) ⋊_{α^{θ},r} H ≅ C_{r}^{∗}(ℤ^{2} ⋊ H, ω̃_{θ})

(and similarly for the full crossed products). Moreover, for all θ ∈ [0, 1], we have graded isomorphisms

K_{∗}(A_{θ} ⋊_{α^{θ},r} H) ≅ K_{∗}(C_{r}^{∗}(ℤ^{2} ⋊ H, ω̃_{θ})) ≅ K_{∗}(C_{r}^{∗}(ℤ^{2} ⋊ H)).

Proof. The isomorphism of C*-algebras follows directly from Lemma 2.1. For the isomorphisms on K-theory, first note that, because SL_{2}(ℤ) satisfies the Baum-Connes Conjecture with coefficients, Theorem 2.5 and Corollary 3.14 of [13] imply that ℤ^{2} ⋊ H does too. Then use Corollary 1.13 and the fact that [ω̃_{θ}] is real in the sense of Definition 1.12. □

Remark 2.3. In our application, to the classification of crossed products A_{θ} ⋊_{α,r} H for certain subgroups H ⊆ SL_{2}(ℤ), we will need the K-groups together with a basis for each. This problem can be simplified substantially, by working with continuous families (in a suitable sense) of projections or unitaries in matrix algebras over the twisted group algebras C_{r}^{∗}(ℤ^{2} ⋊ H, ω̃_{θ}). Thus, for any interval [a, b], consider the cocycle Ω̃: (ℤ^{2} ⋊ H) × (ℤ^{2} ⋊ H) → C([a, b], 𝕋) given by Ω̃( · , ·)(θ) = ω̃_{θ} for θ ∈ [a, b]. By a continuous family, we simply mean a projection (or unitary) in some matrix algebra M_{n}(C([0, 1])⋊_{Ω̃,r} (ℤ^{2} ⋊H)). It follows from Corollary 1.11 that the evaluation map

ev_{θ}: C([0, 1]) ⋊_{Ω̃,r} (ℤ^{2} ⋊ H) → C_{r}^{∗}(ℤ^{2} ⋊ H, ω̃_{θ})

induces an isomorphism in K-theory, and hence it maps a basis to a basis. Lemma 2.1 provides an isomorphism C_{r}^{∗}(ℤ^{2} ⋊ H, ω̃_{θ}) ≅ A_{θ} ⋊_{α^{θ},r} H. An obvious generalization, proved by an easy modification of its proof, provides an isomorphism

C([a, b]) ⋊_{Ω̃,r} (ℤ^{2} ⋊ H) ≅ (C([a, b]) ⋊_{Ω,r} ℤ^{2}) ⋊ H

which moreover respects the evaluation maps ev_{θ} on both algebras. In particular, for projections p_{1}, p_{2}, …, p_{n} in matrix algebras over (C([a, b]) ⋊_{Ω,r} ℤ^{2}) ⋊ H, the following are equivalent:

(1) [p_{1}], [p_{2}], …, [p_{n}] form a basis for K_{0}((C([a, b]) ⋊_{Ω,r} ℤ^{2}) ⋊ H).

(2) For some θ ∈ [a, b], the evaluated classes [ev_{θ}(p_{1})], [ev_{θ}(p_{2})], …, [ev_{θ}(p_{n})]

form a basis for K_{0}(A_{θ} ⋊_{α^{θ},r} H).

(3) For all θ ∈ [a, b], the evaluated classes [ev_{θ}(p_{1})], [ev_{θ}(p_{2})], …, [ev_{θ}(p_{n})]

form a basis for K_{0}(A_{θ} ⋊_{α^{θ},r} H).

Remark 2.4. We should also remark at this point that twisted group algebras of semidirect products Γ = ℤ^{2} ⋊_{A} ℤ, with A a fixed element of SL_{2}(ℤ) and the action of ℤ on ℤ^{2} given by powers of A, were extensively studied by Packer and Raeburn in [53]. They use the embedding of Γ into ℝ^{2} ⋊_{A} ℝ together with Connes’s Thom isomorphism to show that K_{∗}(C^{∗}(Γ, ω)) ≅ K_{∗}(C^{∗}(Γ)) for any cocycle ω ∈ H^{2}(Γ, 𝕋). (See Corollary 2.12 of [53].) Since they also show (Corollary 3.6 of [53]) that every cocycle of such a group Γ is the exponential of a real cocycle, this result also follows from Corollary 1.13 and the proof of Corollary 2.2.

3. The untwisted case

As mentioned above, we are interested in applying the results of the previous section to the case of finite groups F ⊆ SL_{2}(ℤ) which are conjugate to one of the groups ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6} with generators as given in (0.2). In this section we give an explicit computation of the K-theory of the untwisted group algebras C^{∗}(ℤ^{2} ⋊ F). We need some notation.

Notation 3.1. Let G be a discrete group.

(1) For a subgroup H ⊆ G, let ind_{H}^{G}: K_{∗}(C_{r}^{∗}(H)) → K_{∗}(C_{r}^{∗}(G)) be the induction homomorphisms induced by the obvious maps of the reduced group C*-algebras C_{r}^{∗}(H) → C_{r}^{∗}(G).

(2) For a subgroup H ⊆ G such that the index [G : H] is finite, the restriction homomorphism res_{G}^{H}: K_{n}(C_{r}^{∗}(G)) → K_{n}(C_{r}^{∗}(H)) (sometimes written res_{H}^{G}) is given by restriction of scalars: every finitely generated projective C_{r}^{∗}(G)-module is also finitely generated and projective as a module over the subalgebra C_{r}^{∗}(H).

(3) We denote by K̃_{n}(C_{r}^{∗}(G)) the cokernel of the map K_{n}(ℂ) = K_{n}(C^{∗}({1})) → K_{n}(C_{r}^{∗}(G)) induced by the inclusion of the trivial subgroup {1} → G, and we let pr: K_{n}(C_{r}^{∗}(G)) → K̃_{n}(C_{r}^{∗}(G)) be the canonical projection.

The following lemma, known as the double coset formula, will be used several times. It actually holds in greater generality, but all we need here is the version below, which follows from the arguments in the proof of Proposition 5.6(b) in Chapter III of [10].

Lemma 3.2. Suppose that G is a discrete group and H and C are subgroups of G such that C has finite index in G. For g ∈ G, let φ_{g}: C_{r}^{∗}(H) → C_{r}^{∗}(gHg^{−1}) denote the isomorphism induced by conjugation. Then

res_{G}^{C} ∘ ind_{H}^{G} = ∑_{[g]∈C\G/H} (ind_{gHg^{−1}∩C}^{C} ∘ res_{gHg^{−1}}^{gHg^{−1}∩C} ∘ (φ_{g})_{∗})

as maps from K_{0}(C_{r}^{∗}(H)) to K_{0}(C_{r}^{∗}(C)).

Remark 3.3. Let H be a finite group. Then the homomorphism ind_{{1}}^{H}: K_{0}(ℂ) = K_{0}(C^{∗}({1})) → K_{0}(C^{∗}(H)) has a left inverse r: K_{0}(C^{∗}(H)) → K_{0}(ℂ). Letting ℂ carry the trivial representation of H, it sends the class [V] of a finite-dimensional H-representation V to dim_{ℂ}(ℂ ⊗_{C^{∗}(H)} V) · [ℂ]. (The number dim_{ℂ}(ℂ ⊗_{C^{∗}(H)} V) is just the multiplicity of the trivial representation in the decomposition of V into irreducibles.) It induces a splitting

(3.1)  s: K̃_{0}(C^{∗}(H)) → K_{0}(C^{∗}(H))

of pr: K_{0}(C^{∗}(H)) → K̃_{0}(C^{∗}(H)). Namely, s sends x ∈ K̃_{0}(C^{∗}(H)) to y − ( ind_{{1}}^{H} ∘r)(y) for any choice of y ∈ K_{0}(C_{r}^{∗}(G)) with pr(y) = x.

Notation 3.4. Let G be a discrete group, and let H ⊆ G be a finite subgroup. We denote by b_{H} the composition

b_{H}: K̃_{0}(C^{∗}(H)) ⟶^{s} K_{0}(C^{∗}(H)) ⟶^{ind_{H}^{G}} K_{0}(C_{r}^{∗}(G)).

Theorem 3.5. Suppose that F ⊆ SL_{2}(ℤ) is one of the groups ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6} with generators as in (0.2). Then there exist only finitely many conjugacy classes of maximal finite subgroups M ⊆ ℤ^{2} ⋊ F. Let M_{0}, M_{1}, …, M_{l} be a list of representatives for these conjugacy classes. Let b_{j}: K̃_{0}(C^{∗}(M_{j})) → K_{0}(C^{∗}(ℤ^{2} ⋊ F)) for j = 0, 1, …, l be the map b_{M_{j}} of Notation 3.4. Then the following sequence is exact:

0 → K_{0}(C^{∗}({1})) ⊕ (⨁_{j=0}^{l} K̃_{0}(C^{∗}(M_{j}))) ⟶^{ind_{{1}}^{G} ⊕(⨁_{j=0}^{l} b_{j})} K_{0}(C^{∗}(ℤ^{2} ⋊ F))

⟶^{pr ∘ res_{ℤ^{2}⋊F}^{ℤ^{2}}} K̃_{0}(C^{∗}(ℤ^{2})) → 0.

Moreover we have K_{1}(C^{∗}(ℤ^{2} ⋊ F)) = {0} for all F as above.

Remark 3.6. The image of the homomorphism

K_{0}(C^{∗}({1})) ⊕ (⨁_{j=0}^{l} K̃_{0}(C^{∗}(M_{j}))) ⟶^{ind_{{1}}^{G} ⊕(⨁_{j=0}^{l} b_{j})} K_{0}(C^{∗}(ℤ^{2} ⋊ F))

does not depend on the special choice of the section s_{j}: K̃_{0}(C^{∗}(M_{j})) → K_{0}(C^{∗}(M_{j})) in (3.1) which was used in the definition of the maps b_{j}: K̃_{0}(C^{∗}(M_{j})) → K_{0}(C^{∗}(ℤ^{2} ⋊ F)). So the theorem remains true if we replace it by any other section K̃_{0}(C^{∗}(M_{j})) → K_{0}(C^{∗}(M_{j})).

We give two proofs of Theorem 3.5, one depending on the constructions of Sam Walters [72, 75] and some computations from Section 4 below, and the other more topological, although it has much in common with the first. Before doing so, however, we should describe what one can get from known results on the structure of A_{θ} ⋊_{α} F for θ rational. Suppose that θ = p/q in lowest terms, with q > 0. Then the fixed point algebras A_{θ}^{F} and the crossed products A_{θ} ⋊_{α} F have been computed explicitly, in a form from which it is easy to determine the K-theory. In particular, this covers the case C^{∗}(ℤ^{2} ⋊ F). For F = ℤ_{2}, see Theorems 1.2 and 1.3 of [8]; for F = ℤ_{3}, see Theorem 3.4.1 of [29] for the fixed point algebra and the Theorem on page 244 for the crossed product; for F = ℤ_{4}, see Theorem 3.2.5 of [27] for the fixed point algebra and the Theorem 6.2.1 for the crossed product; and for F = ℤ_{6}, see Theorem 3.2.8 of [30] for the fixed point algebra and the Theorem on page 2 for the crossed product. (The proofs for the crossed products are similar to the proofs for the fixed point algebras, and in the cases F = ℤ_{3} and F = ℤ_{6} they are not actually given in the papers.)

We describe briefly what happens for F = ℤ_{6}; the other cases are similar, but sometimes simpler. Identify the algebras (M_{3q})^{2} = M_{3q} ⊕ M_{3q}, (M_{2q})^{3}, and (M_{q})^{6} with algebras of block diagonal matrices in M_{6q}. Choose three distinct points x_{0}, x_{1}, x_{2} ∈ S^{2}. Then

\begin{matrix}A_{p/q} ⋊_{α} ℤ_{6} \\ ≅ \{f ∈ C(S^{2}, M_{6q}): f(x_{0}) ∈ (M_{q})^{6}, f(x_{1}) ∈ (M_{2q})^{3}, and f(x_{2}) ∈ (M_{3q})^{2}\}. \end{matrix} (3.2)

The fixed point algebra has a similar description, in which the functions take values in M_{q}, and the subalgebras (M_{3q})^{2}, (M_{2q})^{3}, and (M_{q})^{6} are replaced by algebras with complicated formulas for the summands and, if q ≤ 6, there are sometimes fewer summands. In particular, if q = 1, then the fixed point algebra is C(S^{2}). Note that the isomorphism class of an algebra given by such a description does not depend on the choice of the points in S^{2} or the particular identifications of the direct sums with subalgebras of M_{6q} or M_{q}.

It follows immediately that 𝕋^{2}/ℤ_{6} ≅ S^{2} (and similarly in the other cases), a fact that will be proved by much less computational methods in the course of the proof of Theorem 3.5. It is also relatively easy to see that, for θ = p/q, we have

(3.3)  K_{0}(A_{θ} ⋊_{α} ℤ_{6}) ≅ ℤ^{10} and K_{1}(A_{θ} ⋊_{α} ℤ_{6}) = 0,

and to write down nine projections in the algebra on the right hand side of (3.2) such that their classes, together with the Bott class, form a basis for K_{0}(A_{p/q} ⋊_{α} ℤ_{6}). (It was this computation for the fixed point algebra, and the similar results for ℤ_{2}, ℤ_{3}, and ℤ_{4}, that led to the conjecture that the fixed point algebras are AF.) It now follows from Remark 2.3, as in the proof of Theorem 4.9, that (3.3) holds for all values of θ. This is enough to make the proof of Theorem 6.3 go through, and conclude that A_{θ} ⋊_{α} ℤ_{6} is an AF algebra. But it does not allow us to decide which AF algebra. Probably the most serious difficulty is that we do not know the trace of the last generator of K_{0}(A_{θ} ⋊_{α} ℤ_{6}) at irrational values of θ. To find it, we presumably must compute, for some rational value of θ (such as θ = 1), the K_{0}-class of the module E_{θ}^{F} of Notation 4.7 in terms of the algebra on the right hand side of (3.2).

First proof of Theorem 3.5. Put G = ℤ^{2} ⋊ F. The group G is amenable. Thus, C^{∗}(G) → C_{r}^{∗}(G) is an isomorphism, and we write C^{∗}(G) throughout. Being amenable, G satisfies the Baum-Connes Conjecture by Corollary 9.2 of [33], that is, the assembly map

μ: K_{n}^{G}(\underline{E}G) ⟶^{≅} K_{n}(C^{∗}(G))

is an isomorphism for all n ∈ ℤ. Let p: K_{0}^{G}(\underline{E}G) → K_{0}(G\\underline{E}G)) be the natural map. The sequence 1 → ℤ^{2} → G → F → 1 has the property that the conjugation action of F on ℤ^{2} is free away from the origin 0 ∈ ℤ^{2}. We now claim that there is an isomorphism

(3.4)  K_{1}(C^{∗}(G)) ≅ K_{1}(G\\underline{E}G)

and an exact sequence

0 → ⨁_{j=1}^{l} K̃_{0}(C^{∗}(M_{j})) \underrightarrow{⨁_{j=0}^{l} b_{j}} K_{0}(C^{∗}(G)) ⟶^{p∘μ^{−1}} K_{0}(G\\underline{E}G) → 0. (3.5)

To get these, apply Theorem 5.1(a) and Remark 5.2 of [18] to the extension 1 → ℤ^{2} → G → F → 1, or apply the more general Theorem 1.6 of [42], taking there K = {1} and G = Q. (Theorem 1.6 of [42] actually gives a long exact sequence, which breaks up into short exact sequences after tensoring with ℚ. However, every group appearing in the long exact sequence is torsion free.)

Clearly G\\underline{E}G is the same as F \𝕋^{2} for the obvious F -action on the twodimensional torus 𝕋^{2} = ℤ^{2}\\underline{E}G = ℤ^{2}\Eℤ^{2}. Since F is a subgroup of SL_{2}(ℤ), its action on 𝕋^{2} is orientation preserving. Also, there are only finitely many points of 𝕋^{2} at which the action is not free. Analyzing these, for example as on page 407 of [68], one can show that F \𝕋^{2} is a compact 2-dimensional manifold. The rational cohomology H^{∗}(G\\underline{E}G; ℚ) agrees with H^{∗}(𝕋^{2}; ℚ)^{F} . Since the action is orientation preserving, F acts trivially on H^{p}(𝕋^{2}; ℚ) for p = 0, 2. Since F acts freely on ℤ^{2} = H_{1}(𝕋^{2}; ℤ) away from {0}, we conclude H^{1}(𝕋^{2}; ℚ)^{F} ≅ Hom_{ℤ}(H_{1}(𝕋^{2}; ℤ)^{F} , ℚ) = \{0\}. Hence G\\underline{E}G = F \𝕋^{2} has the rational cohomology of S^{2} and hence is homeomorphic to S^{2}. This implies that K_{0}(G\\underline{E}G) ≅ ℤ^{2} and K_{1}(G\\underline{E}G) = 0. We conclude from (3.4) that K_{1}(C^{∗}(G)) = 0.

Using the fact that G\\underline{E}G is 2-dimensional, there is an edge homomorphism edge: K_{0}(G\\underline{E}G) → H_{2}(G\\underline{E}G) which comes from the Atiyah-Hirzebruch spectral sequence converging to K_{p+q}(G\\underline{E}G) with E^{2}-term E_{p,q}^{2} = H_{p}(G\\underline{E}G, K_{q}(pt)). (See, for example, Section 13.6 of [76] for the Atiyah-Hirzebruch spectral sequence.) Let f: G\\underline{E}G → pt be the projection onto the one-point space pt . An easy spectral sequence argument shows that

K_{0}(f) ⊕ edge: K_{0}(G\\underline{E}G) ⟶^{≅} K_{0}(pt) ⊕ H_{2}(G\\underline{E}G; ℤ)

is an isomorphism. The composition

K_{0}(C^{∗}(\{1\})) ⟶^{ind_{1}^{G}} K_{0}(C^{∗}(G)) ⟶^{μ^{−1}} K_{0}^{G}(\underline{E}G) ⟶^{p} K_{0}(G\\underline{E}G) ⟶^{f} K_{0}(pt)

is an isomorphism, namely the inverse of the assembly map K_{0}(pt) → K_{0}(ℂ) of the assembly map for the trivial group. Denote by e_{G} the composition

(3.6)  e_{G}: K_{0}(C^{∗}(G)) ⟶^{μ^{−1}} K_{0}^{G}(\underline{E}G) ⟶^{p} K_{0}(G\\underline{E}G) ⟶^{edge} H_{2}(G\\underline{E}G; ℤ).

Now the exact sequence (3.5) yields the exact sequence

(3.7) 0 → K_{0}(C^{∗}({1})) ⊕ ( ⨁_{j=1}^{l} K̃_{0}(C^{∗}(M_{j}))) ⟶^{ind_{{1}}^{G} ⊕(⨁_{j=0}^{l} b_{j})} K_{0}(C^{∗}(G))

⟶^{e_{G}} H_{2}(G\\underline{E}G; ℤ) → 0.

Let res_{G}^{ℤ^{2}} : K_{0}(C^{∗}(G)) → K_{0}(C^{∗}(ℤ^{2})) be the homomorphism induced by restriction to the subgroup of finite index. By the double coset formula of Lemma 3.2 and because each conjugate of each finite group M_{j} has trivial intersection with ℤ^{2} in G, the composition

K_{0}(C^{∗}({1}))⊕(⨁_{j=0}^{l} K̃_{0}(C^{∗}(M_{j}))) ⟶^{ind_{{1}}^{G} ⊕(⨁_{j=0}^{l} b_{j})} K_{0}(C^{∗}(G)) ⟶^{res_{G}^{ℤ^{2}}} K_{0}(C^{∗}(ℤ^{2}))

factors through the map ind_{{1}}^{ℤ^{2}} : K_{0}(C^{∗}({1})) → K_{0}(C^{∗}(ℤ^{2})). Combining this with the exact sequence (3.7) yields a commutative diagram

0  0
↓  ↓
K_{0}(C^{∗}({1})) ⊕ (⨁_{j=0}^{l} K̃_{0}(C^{∗}(M_{j}))) ⟶^{id} K_{0}(C^{∗}({1})) ⊕ (⨁_{j=0}^{l} K̃_{0}(C^{∗}(M_{j})))
ind_{{1}}^{G} ⊕(⨁_{j=0}^{l} b_{j})↓  ↓^{ind_{{1}}^{G} ⊕(⨁_{j=0}^{l} b_{j})}
K_{0}(C^{∗}(G))  ⟶^{id}  K_{0}(C^{∗}(G))
e_{G}↓  ↓_{pr ∘ res_{G}^{ℤ^{2}}}
H_{2}(G\\underline{E}G; ℤ)  ⟶^{ω}  K̃_{0}(C^{∗}(ℤ^{2}))
↓  ↓
0  0

for an appropriate map ω. Lemma 4.8 below (based on Sam Walters’ construction of Fourier modules [72, 75]; the proof does not depend on Theorem 3.5) provides explicit elements in K_{0}(C^{∗}(G)) which are mapped to a generator in K̃_{0}(C^{∗}(ℤ^{2})) under pr ∘ res_{G}^{ℤ^{2}} . This implies that pr ∘ res_{G}^{ℤ^{2}} is surjective. Hence ω is a surjection of infinite cyclic groups and therefore an isomorphism. Since the left column in the diagram above is exact, the same is true for the right column. □

We now give a second proof, not relying on Lemma 4.8. Parts of the argument are the same as in the first proof, and we refer freely to it.

Second proof of Theorem 3.5. Analogously to the definition of e_{G} in (3.6), we define a surjection

e_{ℤ^{2}} : K_{0}(C^{∗}(ℤ^{2})) → H_{2}(ℤ^{2}\Eℤ^{2})

for the subgroup ℤ^{2} ⊆ G. Let q: 𝕋^{2} = ℤ^{2}\\underline{E}ℤ^{2} → F \𝕋^{2} = G\\underline{E}G be the projection. Let i: ℤ^{2} → G be the inclusion. Then the following diagram commutes:

\begin{matrix}K_{0}(C^{∗}(ℤ^{2})) & ⟶^{e_{ℤ^{2}}} & H_{2}(𝕋^{2}; ℤ) \\ ^{K_{0}(i)}↓ & & ↓^{H_{2}(q)} \\ K_{0}(C^{∗}(G)) & ⟶_{e_{G}} & H_{2}(F \𝕋^{2}; ℤ). \end{matrix} (3.8)

The F -action on 𝕋^{2} is orientation preserving and has at least one free orbit. The quotient F \𝕋^{2} is a two dimensional manifold as in the first proof, so q: 𝕋^{2} → F \𝕋^{2} is a map of closed oriented surfaces of degree card(F) and thus induces an injective map H_{2}(q) of infinite cyclic groups whose image has index card(F).

An easy spectral sequence argument shows that the image of

ind_{{1}}^{ℤ^{2}} : K_{0}(C^{∗}({1})) → K_{0}(C^{∗}(ℤ^{2}))

is the kernel of the surjection e_{ℤ^{2}} : K_{0}(C^{∗}(ℤ^{2})) → H_{2}(𝕋^{2}; ℤ). Hence we obtain from (3.7) a map μ: H_{2}(F \𝕋^{2}; ℤ) → H_{2}(𝕋^{2}; ℤ) such that the following square commutes:

\begin{matrix}K_{0}(C^{∗}(G)) & ⟶^{e_{G}} & H_{2}(F \𝕋^{2}; ℤ) \\ _{res}_{G}^{ℤ^{2}} ↓ & & ↓^{μ} \\ K_{0}(C^{∗}(ℤ^{2})) & ⟶_{e_{ℤ^{2}}} & H_{2}(𝕋^{2}; ℤ), \end{matrix} (3.9)

and an isomorphism

ν: H_{2}(𝕋^{2}; ℤ) ⟶^{≅} K̃_{0}(C^{∗}(ℤ^{2}))

such that the composition ν ∘ e_{ℤ^{2}} agrees with the projection pr: K_{0}(C^{∗}(ℤ^{2})) → K̃_{0}(C^{∗}(ℤ^{2})). Hence the following diagram commutes:

\begin{matrix}K_{0}(C^{∗}(G)) & ⟶^{e_{G}} & H_{2}(F \𝕋^{2}; ℤ) \\ _{res}_{G}^{ℤ^{2}} ↓ & & ↓^{ν∘μ} \\ K_{0}(C^{∗}(ℤ^{2})) & ⟶_{pr} & K̃_{0}(C^{∗}(ℤ^{2})). \end{matrix} (3.10)

In view of the exact sequence (3.7) it remains to show that μ is bijective. Again by the double coset formula (Lemma 3.2), the composition

K_{0}(C^{∗}(ℤ^{2})) ⟶^{K_{0}(i)} K_{0}(C^{∗}(G)) ⟶^{res_{G}^{ℤ^{2}}} K_{0}(C^{∗}(ℤ^{2}))

is ∑_{j=1}^{card(F)} K_{0}(σ)^{j}, where σ: ℤ^{2} → ℤ^{2} is given by multiplication by the generator of the finite cyclic group F. The map σ induces an orientation preserving automorphism of 𝕋^{2} and hence the identity H_{2}(𝕋^{2}). Therefore the following diagram commutes:

K_{0}(C^{∗}(ℤ^{2})) ⟶^{e_{ℤ^{2}}} H_{2}(𝕋^{2}; ℤ)
res_{G}^{ℤ^{2}} ∘K_{0}(i)↓  ↓^{card(F)·id}
K_{0}(C^{∗}(ℤ^{2})) ⟶_{e_{ℤ^{2}}} H_{2}(𝕋^{2}; ℤ).

(3.11)

Fix a generator x of the infinite cyclic group H_{2}(𝕋^{2}; ℤ). Choose y ∈ K_{0}(C^{∗}(ℤ^{2})) with e_{ℤ^{2}}(y) = x. Let z ∈ K_{0}(C^{∗}(G)) be the image of y under the homomorphism K_{0}(i): K_{0}(C^{∗}(ℤ^{2})) → K_{0}(C^{∗}(G)). We conclude from (3.8) and the fact that H_{2}(pr) is injective and has cokernel of order card(F) that e_{G}(z) is card(F)·a for a suitable generator a of the infinite cyclic group H_{2}(F \𝕋^{2}; ℤ). We get from (3.9) and (3.11) that

card(F) · μ(a) = μ ∘ e_{G}(z) = card(F) · x.

Since a and x are generators of infinite cyclic groups, μ is bijective. (Notice that the map ω agrees with the map ν ∘ μ in the first proof.) This finishes the second proof of Theorem 3.5. □

It is clear how to use Theorem 3.5 to give an explicit basis for K_{0}(ℤ^{2} ⋊ F), and we do this in all four cases. Note first that K_{0}(C^{∗}(ℤ_{k})) is the free abelian group of rank k. If t is a generator for ℤ_{k}, identified with its canonical image in C^{∗}(ℤ_{k}), and ζ = e^{2πi/k}, then an explicit basis is given by the classes of the projections

p_{j} = \frac{1}{k} ∑_{l=0}^{k−1} (ζ^{j}t)^{l}

for j = 0, …, k − 1. One easily checks that ∑_{j=0}^{k−1} p_{j} = 1 ∈ C^{∗}(ℤ_{k}), and therefore we may replace any one of the above projections by [1] to obtain a new basis. In particular, it follows that K̃_{0}(C^{∗}(ℤ_{k})), as defined in Notation 3.1(3), is a free abelian group of rank k − 1 with basis

([p_{0}], [p_{1}], …, [p_{k−2}]).

In the following examples we give a complete set of representatives of the conjugacy classes of maximal finite subgroups. The elementary calculation that this is indeed such a system is carried out for F = ℤ_{4} in Lemma 2.2 of [42]. The analogous calculations in the other cases are left to the reader.

Example 3.7. (a) The case F = ℤ_{2}. The group ℤ^{2} ⋊ ℤ_{2} has the presentation

ℤ^{2} ⋊ ℤ_{2} = ⟨u, v, t: t^{2} = 1, uv = vu, tut = u^{−1}, tvt = v^{−1}⟩.

The maximal finite subgroups are represented, up to conjugacy, by the four groups

M_{0} = ⟨t⟩, M_{1} = ⟨ut⟩, M_{2} = ⟨vt⟩, M_{3} = ⟨uvt⟩,

which are all isomorphic to ℤ_{2}. Hence it follows from Theorem 3.5 that, for any S ∈ K_{0}(C^{∗}(ℤ^{2} ⋊ℤ_{2})) which maps onto a generator of K̃_{0}(C^{∗}(ℤ^{2})) ≅ ℤ via pr ∘ res_{ℤ^{2}⋊ℤ_{2}}^{ℤ^{2}},

([1], [\frac{1}{2}(1 + t)], [\frac{1}{2}(1 + ut)], [\frac{1}{2}(1 + vt)], [\frac{1}{2}(1 + uvt)], S)

is a basis for K_{0}(C^{∗}(ℤ^{2} ⋊ ℤ_{2})) ≅ ℤ^{6}.

(b) The case F = ℤ_{3}. Here we have the presentation

ℤ^{2} ⋊ ℤ_{3} = ⟨u, v, t: t^{3} = 1, uv = vu, tut^{−1} = u^{−1}v, tvt^{−1} = u^{−1}⟩.

The conjugacy classes of the maximal finite subgroups are represented by

M_{0} = ⟨t⟩, M_{1} = ⟨ut⟩, M_{2} = ⟨u^{2}t⟩,

which are all isomorphic to ℤ_{3}. Set ζ = \frac{1}{2}( − 1 + i\sqrt{3}), and define

p_{0} = \frac{1}{3}(1 + t + t^{2}), q_{0} = \frac{1}{3}(1 + ut + (ut)^{2}), r_{0} = \frac{1}{3}(1 + u^{2}t + (u^{2}t)^{2}),

p_{1} = \frac{1}{3}(1 + ζt + (ζt)^{2}), q_{1} = \frac{1}{3}(1 + ζut + (ζut)^{2}), r_{1} = \frac{1}{3}(1 + ζu^{2}t + (ζu^{2}t)^{2}).

Then, if S ∈ K_{0}(C^{∗}(ℤ^{2} ⋊ ℤ_{3})) maps to a generator of K̃_{0}(C^{∗}(ℤ^{2})) ≅ ℤ, we obtain from Theorem 3.5 that

([1], [p_{0}], [p_{1}], [q_{0}], [q_{1}], [r_{0}], [r_{1}], S)

is a basis for K_{0}(C^{∗}(ℤ^{2} ⋊ ℤ_{3})) ≅ ℤ^{8}.

(c) The case F = ℤ_{4}. In this case we have the presentation

ℤ^{2} ⋊ ℤ_{4} = ⟨u, v, t: t^{4} = 1, uv = vu, tut^{−1} = v, tvt^{−1} = u^{−1}⟩.

The maximal finite subgroups are given up to conjugacy by

M_{0} = ⟨t⟩, M_{1} = ⟨ut⟩, M_{2} = ⟨ut^{2}⟩.

The groups M_{0} and M_{1} are isomorphic to ℤ_{4}, and M_{2} ≅ ℤ_{2}. Define

p_{0} = \frac{1}{4}(1 + t + t^{2} + t^{3}), p_{1} = \frac{1}{4}(1 + it − t^{2} − it^{3}), p_{2} = \frac{1}{4}(1 − t + t^{2} − t^{3}),

q_{0} = \frac{1}{4}(1 + ut + (ut)^{2} + (ut)^{3}), q_{1} = \frac{1}{4}(1 + iut − (ut)^{2} − i(ut)^{3}),
q_{2} = \frac{1}{4}(1 − ut + (ut)^{2} − (ut)^{3}), and r = \frac{1}{2}(1 + ut^{2}).

Then, if S ∈ K_{0}(C^{∗}(ℤ^{2} ⋊ ℤ_{4})) maps to a generator of K̃_{0}(C^{∗}(ℤ^{2})) ≅ ℤ, we obtain from Theorem 3.5 the basis

([1], [p_{0}], [p_{1}], [p_{2}], [q_{0}], [q_{1}], [q_{2}], [r], S)

of K_{0}(C^{∗}(ℤ^{2} ⋊ ℤ_{4})) ≅ ℤ^{9}.

(d) The case F = ℤ_{6}. Here we have the presentation

ℤ^{2} ⋊ ℤ_{6} = ⟨u, v, t: t^{6} = 1, uv = vu, tut^{−1} = v, tvt^{−1} = u^{−1}v⟩.

The maximal finite subgroups are given up to conjugacy by

M_{0} = ⟨t⟩, M_{1} = ⟨ut^{2}⟩, M_{2} = ⟨ut^{3}⟩,

with M_{0} ≅ ℤ_{6}, M_{1} ≅ ℤ_{3}, and M_{2} ≅ ℤ_{2}. Set ζ = \frac{1}{2}(1 + i\sqrt{3}), define

p_{j} = \frac{1}{6}(1 + (ζ^{j}t) + (ζ^{j}t)^{2} + (ζ^{j}t)^{3} + (ζ^{j}t)^{4} + (ζ^{j}t)^{5})

for j = 0, 1, 2, 3, 4, and define

q_{0} = \frac{1}{3}(1 + ut^{2} + (ut^{2})^{2}), q_{1} = \frac{1}{3}(1 + ζ^{2}ut^{2} + (ζ^{2}ut^{2})^{2}), and r = \frac{1}{2}(1 + ut^{3}).

Then, if S ∈ K_{0}(C^{∗}(ℤ^{2} ⋊ ℤ_{6})) maps to a generator of K̃_{0}(C^{∗}(ℤ^{2})) ≅ ℤ, we obtain from Theorem 3.5 that

([1], [p_{0}], [p_{1}], [p_{2}], [p_{3}], [p_{4}], [q_{0}], [q_{1}], [r], S)

is a basis of K_{0}(C^{∗}(ℤ^{2} ⋊ ℤ_{6})) ≅ ℤ^{10}.

Using the construction of the Fourier modules due to Sam Walters (see [72, 75]) we later explicitly construct candidates for S in all four cases considered above.

As a direct consequence of Example 3.7 and Corollary 2.2 we now get:

Corollary 3.8. Let F ⊆ SL_{2}(ℤ) be one of the groups ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6} with generators as described above and let F act on the rotation algebra A_{θ} for θ ∈ [0, 1] via the restriction of the action of SL_{2}(ℤ) as described in (0.1) and (0.2). Then K_{1}(A_{θ} ⋊_{α} F) = 0 for all F and θ, and for all θ ∈ [0, 1] we have

K_{0}(A_{θ} ⋊_{α} ℤ_{2}) ≅ ℤ^{6}, K_{0}(A_{θ} ⋊_{α} ℤ_{3}) ≅ ℤ^{8},
K_{0}(A_{θ} ⋊_{α} ℤ_{4}) ≅ ℤ^{9}, and K_{0}(A_{θ} ⋊_{α} ℤ_{6}) ≅ ℤ^{10}.

4. A basis for K_{0}(A_{θ} ⋊_{α} F)

Let θ ∈ [0, 1]. In this section we present an explicit basis for K_{0}(A_{θ} ⋊_{α} F) for F = ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6}.

Notation 4.1. To simplify notation we write from now on e(x) = e^{2πix} for all x ∈ ℝ.

Recall from Corollary 2.2 that A_{θ} ⋊_{α} F ≅ C^{∗}(ℤ^{2} ⋊ F, ω̃_{θ}).

Lemma 4.2. Let θ ∈ [0, 1], and let F be one of ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6}. Then A_{θ} ⋊_{α} F is the universal C*-algebra generated by three unitaries

u_{θ}, v_{θ}, t_{θ},

subject to the relations v_{θ}u_{θ} = e(θ)u_{θ}v_{θ} and

for F = ℤ_{2}:  t_{θ}^{2} = t_{θ}, t_{θ}u_{θ}t_{θ}^{−1} = u_{θ}^{−1},  t_{θ}v_{θ}t_{θ}^{−1} = v_{θ}^{−1}
for F = ℤ_{3}:  t_{θ}^{3} = t_{θ}, t_{θ}u_{θ}t_{θ}^{−1} = e( − \frac{1}{2}θ)u_{θ}^{−1}v_{θ}, t_{θ}v_{θ}t_{θ}^{−1} = u_{θ}^{−1}
for F = ℤ_{4}:  t_{θ}^{4} = t_{θ}, t_{θ}u_{θ}t_{θ}^{−1} = v_{θ},  t_{θ}v_{θ}t_{θ}^{−1} = u_{θ}^{−1}
for F = ℤ_{6}:  t_{θ}^{6} = t_{θ}, t_{θ}u_{θ}t_{θ}^{−1} = v_{θ},  t_{θ}v_{θ}t_{θ}^{−1} = e( − \frac{1}{2}θ)u_{θ}^{−1}v_{θ}.

The C*-algebra generated by u_{θ} and v_{θ} is A_{θ}. Moreover, the homomorphism given on the generators by

u_{0} ↦ u, v_{0} ↦ v and t_{0} ↦ t

is an isomorphism A_{0} ⋊_{α} F → C^{∗}(ℤ^{2} ⋊ F), and the homomorphism given on the generators by

u_{1} ↦ −u, v_{1} ↦ −v and t_{1} ↦ t

is an isomorphism A_{1} ⋊_{α} F → C^{∗}(ℤ^{2} ⋊ F).

Proof. The first part follows from the description of the action of SL_{2}(ℤ) on A_{θ} as given in Corollary 2.2 and the realization of the groups F = ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6} ⊆ SL_{2}(ℤ) with generators as in (0.2). The last part is trivial for θ = 0, and follows from e( − \frac{1}{2}θ) = −1 for θ = 1. □

Although the cocycle ω̃_{1} is in the class of the trivial cocycle, it is not trivial itself, so it has to be taken into account in the computations below!

We now fix some a ∈ (0, 1]. Consider the 2-cocycle

Ω̃_{a}: (ℤ^{2} ⋊ F) × (ℤ^{2} ⋊ F) → C([a, 1], 𝕋)

given by Ω̃_{a}( · , ·, θ) = ω̃_{θ}( · , ·) for θ ∈ [a, 1]. It is clear that it restricts at θ to the cocycle ω_{θ} on ℤ^{2}. For convenience, we write C^{∗}(ℤ^{2} ⋊ F, Ω̃_{a}) for the twisted crossed product C([a, 1]) ⋊Ω̃_{a} (ℤ^{2} ⋊ F) and C^{∗}(ℤ^{2}, Ω_{a}) for the twisted crossed product C([a, 1]) ⋊_{Ω_{a}} ℤ^{2}. As in Remark 2.3, we may write

(4.1)  C^{∗}(ℤ^{2} ⋊ F, Ω̃_{a}) ≅ C^{∗}(ℤ^{2}, Ω_{a}) ⋊ F

where the action of F on C^{∗}(ℤ^{2}, Ω_{a}) is given fiberwise by the actions on the fibers A_{θ} = C^{∗}(ℤ^{2}, ω_{θ}) for θ ∈ [a, 1].

We want to construct suitable elements of K_{0}(C^{∗}(ℤ^{2} ⋊ F, Ω̃_{a})) which, with respect to the identification C^{∗}(ℤ^{2} ⋊ F, ω̃_{1}) ≅ C^{∗}(ℤ^{2} ⋊ F) of Lemma 4.2, project to classes in the fiber at 1 as given in Example 3.7.

Notation 4.3. Let a ∈ (0, 1]. Since G = ℤ^{2} ⋊ F is discrete, there is a canonical map from ℤ^{2} ⋊F into the group of unitaries of C^{∗}(ℤ^{2} ⋊F, Ω̃_{a}) which sends a group element to the corresponding Dirac function. Let U_{a}, V_{a}, T_{a} ∈ C^{∗}(ℤ^{2} ⋊ F, Ω̃_{a}) be the images of the generators u, v, t ∈ ℤ^{2} ⋊ F under this map. We use the same notation for these unitaries regarded as elements of suitable dense subalgebras, and, for U_{a} and V_{a}, with C^{∗}(ℤ^{2}, Ω_{a}) in place of C^{∗}(ℤ^{2} ⋊ F, Ω̃_{a}).

These unitaries clearly project to the generators u_{θ}, v_{θ}, t_{θ} ∈ C^{∗}(ℤ^{2} ⋊ F, ω̃_{θ}) for θ ∈ [a, 1]. Moreover, every function φ ∈ C([a, 1]) can be considered canonically as a central element of C^{∗}(ℤ^{2} ⋊ F, Ω̃_{a}) by identifying φ with δ_{1} ⊗ φ ∈ C_{c}(ℤ^{2} ⋊ F, C([a, 1])) ⊆ C^{∗}(ℤ^{2} ⋊ F, Ω̃_{a}) (where 1 denotes the identity of ℤ^{2} ⋊ F). In the lemma below, we identify the generators u, v, t of C^{∗}(ℤ^{2} ⋊ F) with the generators −u_{1}, −v_{1}, t_{1} of C^{∗}(ℤ^{2} ⋊ F, ω̃_{1}) as in the last part of Lemma 4.2.

Lemma 4.4. Recalling Notation 4.1, define functions φ_{1}, φ_{2}, φ_{3} ∈ C([a, 1]) by

φ_{1}(θ) = −e(\frac{1}{2}θ), φ_{2}(θ) = e(\frac{1}{6}(2 + θ)), and φ_{3}(θ) = ie(\frac{1}{4}θ).

Then:

(1) If F = ℤ_{2}, the elements T_{a}, −U_{a}T_{a}, −V_{a}T_{a}, and φ_{1}U_{a}V_{a}T_{a} are unitaries of order two which project onto the elements t, ut, vt, and uvt in the fiber at 1.

(2) If F = ℤ_{3}, the elements T_{a}, φ_{2}U_{a}T_{a}, and U_{a}^{2}T_{a} are unitaries of order three which project onto the elements t, ut, and u^{2}t in the fiber at 1.

(3) If F = ℤ_{4}, the elements T_{a}, and φ_{3}U_{a}T_{a} are unitaries of order four and −U_{a}T_{a}^{2} is a unitary of order two which project onto the elements t, ut, and ut^{2} in the fiber at 1.

(4) If F = ℤ_{6}, then T_{a} is a unitary of order six, φ_{2}U_{a}T_{a}^{2} is a unitary of order three, and −U_{a}T_{a}^{3} is a unitary of order two, and they project onto the elements t, ut^{2}, and ut^{3} in the fiber at 1.

Proof. The proof is a straight-forward (but tedious) computation using the relations for the unitaries u_{θ}, v_{θ}, t_{θ} in each fiber C^{∗}(ℤ^{2} ⋊ F, ω̃_{θ}). We leave the details for the reader. □

As a consequence of Lemma 4.4, we can now use exactly the same formulas as for the explicit projections in C^{∗}(ℤ^{2} ⋊ F) presented in Example 3.7, replacing the combinations of u, v, t in the example by appropriate combinations of U_{a}, V_{a}, T_{a}, to obtain projections in C^{∗}(ℤ^{2} ⋊ F, Ω̃_{a}) which project onto the ones in C^{∗}(ℤ^{2} ⋊ F) ≅ C^{∗}(ℤ^{2} ⋊ F, ω̃_{1}) via evaluation at 1. Therefore, all we have to do is to find a class S_{a} ∈ K_{0}(C^{∗}(ℤ^{2} ⋊ F, Ω̃_{a})) which projects onto a class S ∈ K_{0}(C^{∗}(ℤ^{2} ⋊ F)) via evaluation at 1 with the properties as in Example 3.7. Before we construct this element, it is convenient to recall some basic properties about equivariant K-theory of C*-algebras with respect to finite group actions.

Proposition 4.5. Suppose that F is a finite group and α: F → Aut(A) is an action of F on the C*-algebra A. Suppose further that E is a finitely generated projective (right) A-module together with a right action W: F → Aut(E), written (ξ, g) ↦ ξW_{g}, such that (ξW_{g})a = (ξα_{g}(a))W_{g} for all ξ ∈ E, a ∈ A, and g ∈ F. Then E becomes a finitely generated projective A ⋊_{α} F -module with action defined by

ξ · ( ∑_{g∈F} a_{g}δ_{g}) = ∑_{g∈F}(ξa_{g})W_{g}.

Moreover, the restriction to A of this module structure is just the original A-module E, with the action of F forgotten.

Proof. This is the definition of the map in the Green-Julg theorem (Theorem 11.7.1 of [5] or Theorem 2.6.1 of [56]). The last statement is obvious. □

We will use Proposition 4.5 and the isomorphism (4.1) to construct finitely generated projective C^{∗}(ℤ^{2} ⋊ F, Ω̃_{a})-modules which represent a suitable class S_{a}. To do this, we exploit below some basic constructions and ideas of Alain Connes and Marc Rieffel in [16, 63] and Sam Walters in [72, 75]. Define a new cocycle Ω_{1/a} ∈ Z^{2}(ℤ^{2}, C([a, 1], 𝕋)) by setting Ω_{1/a}( · , ·)(θ) = ω_{1/θ} for all θ ∈ [a, 1]. Set A = C^{∗}(ℤ^{2}, Ω_{a}) and B = C^{∗}(ℤ^{2}, Ω_{1/a}). Then the fiber B_{θ} of B at θ ∈ [a, 1] is the rotation algebra A_{1/θ} = C^{∗}(ℤ^{2}, ω_{1/θ}). Consider the space S[a, 1] consisting of all complex functions on ℝ×[a, 1] which are smooth and rapidly decreasing in the first variable and continuous in the second variable in each derivative of the first variable. Denote by S(ℤ^{2}, Ω_{a}) the set of rapidly decreasing C([a, 1])-valued functions on ℤ^{2}, viewed as a (dense) subalgebra of C^{∗}(ℤ^{2}, Ω_{a}), and denote by S(ℤ^{2}, Ω_{1/a}) the same set, viewed as a (dense) subalgebra of C^{∗}(ℤ^{2}, Ω_{1/a}). With U_{a} and V_{a} as in Notation 4.3, we may represent each element φ ∈ S(ℤ^{2}, Ω_{a}) as an infinite sum

(4.2)  φ = ∑_{n,m∈ℤ} β_{n,m} U_{a}^{n}V_{a}^{m}

with β_{n,m} ∈ C([a, 1]) ⊆ Z(C^{∗}(ℤ^{2}, Ω_{a})) and ( \begin{matrix}n \\ m \end{matrix}) → ∥β_{n,m}∥_{∞} rapidly decreasing on ℤ^{2}. Similarly, we may write each element ψ ∈ S(ℤ^{2}, Ω_{1/a}) as a sum

(4.3)  ψ = ∑_{n,m∈ℤ} α_{n,m} U_{1/a}^{n}V_{1/a}^{m}

with α_{n,m} ∈ C([a, 1]). We define a right action of S(ℤ^{2}, Ω_{a}) on S[a, 1] via the actions of U_{a}, V_{a}, and f ∈ C([a, 1]) given by (4.4) (ξ·U_{a})(s, θ) = ξ(s+θ, θ), (ξ·V_{a})(s, θ) = e(s)ξ(s, θ), and (ξ·f)(s, θ) = f(θ)ξ(s, θ).

Moreover, we define an S(ℤ^{2}, Ω_{a})-valued inner product on S[a, 1] as follows. For n, m ∈ ℤ, set

(4.5)  ⟨ξ, η⟩_{n,m}^{a}(θ) = θ ∫_{−∞}^{∞} \overline{ξ(x + nθ, θ)}η(x, θ)e(−mx) dx.

Then set

(4.6)  ⟨ξ, η⟩_{S(ℤ^{2},Ω_{a})} = ∑_{n,m∈ℤ}⟨ξ, η⟩_{n,m}^{a}U_{a}^{n}V_{a}^{m}.

There is also a left action of S(ℤ^{2}, Ω_{1/a}) on S[a, 1] given by (4.7) U_{1/a}·ξ(s, θ) = ξ(s+1, θ), V_{1/a}·ξ(s, θ) = e(−s/θ)ξ(s, θ), α·ξ(s, θ) = α(θ)ξ(s, θ), for α ∈ C([a, 1]), and a left inner product

(4.8)  _{S(ℤ^{2}, Ω_{1/a})}⟨ξ, η⟩ = ∑_{n,m∈ℤ}⟨ξ, η⟩_{n,m}^{1/a}U_{1/a}^{n}V_{1/a}^{m}

with

⟨ξ, η⟩_{n,m}^{1/a}(θ) = ∫_{−∞}^{∞} ξ(x − n, θ)\overline{η(x, θ)}e(mx/θ) dx.

Evaluated at each fixed θ ∈ [a, 1], these actions and inner products determine B_{θ} − A_{θ} imprimitivity bimodules E_{θ} = \overline{S(ℝ)} as in [63] (Theorem 1.1 and the proof of Lemma 1.5). The formulas given here are from the beginning of Section 3 of [72]. (The paper [63] uses a dense subalgebra of A_{θ} consisting of functions on ℤ × 𝕋.) It follows then easily from the fact that all operations are C([a, 1])-linear that S[a, 1] completes to give a B −A imprimitivity bimodule E. Since both B and A are unital, it follows from the arguments given after Notation 1.3 of [63] that E is a finitely generated projective A-module with respect to the given action of A on E. In order to construct a module over C^{∗}(ℤ^{2} ⋊ F, Ω̃_{a}) = A ⋊ F, it thus suffices to construct a suitable action of F on E. The operator W_{t} in the following proposition is really the action of the unitary T_{a} on the module.

Proposition 4.6. Let F be one of the groups ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6} with fixed generator t. We define an action of F on the dense subspace S[a, 1] (described above) of the B − A imprimitivity bimodule E via the action of the generator t given by:

(ξW_{t})(s, θ) = ξ(−s, θ)  (for F = ℤ_{2})
(ξW_{t})(s, θ) = e^{−πi/12}θ^{−1/2}e(s^{2}/(2θ)) ∫_{−∞}^{∞} ξ(x, θ)e(sx/θ) dx  (for F = ℤ_{3})
(ξW_{t})(s, θ) = θ^{−1/2} ∫_{−∞}^{∞} ξ(x, θ)e(sx/θ) dx  (for F = ℤ_{4})
(ξW_{t})(s, θ) = e^{πi/12}θ^{−1/2} ∫_{−∞}^{∞} ξ(x, θ)e((2sx − x^{2})/(2θ)) dx  (for F = ℤ_{6}).

Then these actions extend to actions on E such that E becomes an F -equivariant finitely generated projective A-module as in Proposition 4.5.

Proof. The operator which appears in the case F = ℤ_{6} is, in the fiber over θ ∈ [a, 1], the hexic transform of Walters [75] with respect to the parameter μ = \frac{1}{2θ} . It follows from Theorem 1 of [75] that it has period six and that its square is the operator which appears in the case F = ℤ_{3}, which therefore has period three. The operator which appears in case F = ℤ_{4} is, in the fiber over θ, the one introduced by Walters in Section 3 of [72], where it is shown that it has period four with square equal to the operator which appears in case F = ℤ_{2}. We have to show that, in all four cases, there exists a unique extension of W_{t} to E. This will be clear as soon as we verify the equation

⟨ξW_{t}, ηW_{t}⟩_{A} = α_{t^{−1}}(⟨ξ, η⟩_{A})

for all ξ, η ∈ S[a, 1] and in all cases F = ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6}. The cases F = ℤ_{2} and F = ℤ_{3} will follow directly from the cases F = ℤ_{4} and F = ℤ_{6}. As a sample, we do the details for the case F = ℤ_{6} here and leave the computation for the case F = ℤ_{4} to the reader. (Note that these relations in the case F = ℤ_{4} have been used implicitly in Section 3 of [72].)

Replacing ξ by ξW_{t}^{−1} and applying α_{t} on both sides, we reduce to checking the equation

(4.9)  ⟨ξW_{t}^{−1}, η⟩_{A} = α_{t}(⟨ξ, ηW_{t}⟩_{A}).

The formula for the transformation ξ ↦ ξW_{t}^{−1} is given in [75] (see the remark immediately after Theorem 1 of [75], and continue to take μ = \frac{1}{2θ}), and is

(4.10)  (ξW_{t}^{−1})(s, θ) = \frac{e^{−πi/12}}{\sqrt{θ}} e(\frac{1}{2}θ^{−1}s^{2}) ∫_{−∞}^{∞} ξ(x, θ)e(−sx/θ) dx.

Let φ ∈ S(ℤ^{2}, Ω_{a}) be given φ = ∑_{n,m∈ℤ} β_{n,m} U_{a}^{n}V_{a}^{m} as in (4.2). We use the definition of the action α_{t} (for example, see the commutation relations in Lemma 4.2 for the case F = ℤ_{6}) and, at the third step, the commutation relation v_{θ}u_{θ}^{−1} = e(−θ)u_{θ}^{−1}v_{θ}, to get, for θ ∈ [a, 1],

(α_{t}(φ))(θ) = ∑_{n,m∈ℤ} β_{n,m}(θ)α_{t}^{θ}(u_{θ})^{n}α_{t}^{θ}(v_{θ})^{m}

= ∑_{n,m} β_{n,m}(θ)e( − \frac{1}{2}mθ)v_{θ}^{n}(u_{θ}^{−1}v_{θ})^{m}

= ∑_{n,m} β_{n,m}(θ)e( − \frac{1}{2}θ(m^{2} + 2nm))u_{θ}^{−m}v_{θ}^{n+m}

= ∑_{k,l} β_{l+k, −k}(θ)e(\frac{1}{2}θ(2k(l + k) − k^{2}))u_{θ}^{k}v_{θ}^{l}.

Using this formula together with (4.6), we see that we have to check the identity

(4.11)  ⟨ξW_{t}^{−1}, η⟩_{k,l}^{a}(θ) = ⟨ξ, ηW_{t}⟩_{k+l, −k}^{a}(θ)e(\frac{1}{2}θ[2k(k + l) − k^{2}])

for all k, l ∈ ℤ. Combining (4.5) with (4.10) gives for the left hand side:

\begin{matrix}⟨ξW_{t}^{−1}, η⟩_{k,l}^{a}(θ) \\ = e^{πi/12}\sqrt{θ} ∫_{−∞}^{∞} ∫_{−∞}^{∞} \overline{ξ(r, θ)}η(x, θ)e(\frac{1}{2}θ^{−1}[2(x + kθ)r − (x + kθ)^{2} − 2θlx]) dr dx. \end{matrix} (4.12)

For the right hand side of (4.11), first define

\begin{matrix}ψ(r, x, θ, k, l) \\ = e (\frac{1}{2θ} [2(r − (l + k)θ)x − x^{2} + 2θk(r − (l + k)θ) + θ^{2}(2k(k + l) − k^{2})]) . \end{matrix}

Calculate, using at the first step (4.5) and the formula for W_{t} as given in the proposition, and using at the second step the change of variables r to r − (k + l)θ, to get:

⟨ξ, ηW_{t}⟩_{k+l, −k}^{a}(θ)e(\frac{1}{2}θ[2k(k + l) − k^{2}])

= e^{πi/12}\sqrt{θ} ∫_{−∞}^{∞} ∫_{−∞}^{∞} \overline{ξ(r + (k + l)θ, θ)}η(x, θ)

e(\frac{1}{2}θ^{−1}(2rx − x^{2} + 2θkr))e(\frac{1}{2}θ[2k(l + k) − k^{2}]) dr dx

= e^{πi/12}\sqrt{θ} ∫_{−∞}^{∞} ∫_{−∞}^{∞} \overline{ξ(r, θ)}η(x, θ)ψ(r, x, θ, k, l) dr dx.

So the result follows from the identity

2(x + kθ)r − (x + kθ)^{2} − 2θlx

= 2(r − (l + k)θ)x − x^{2} + 2θk(r − (l + k)θ) + θ^{2}(2k(k + l) − k^{2}).

We still have to check the equation

(ξW_{t})φ = (ξα_{t}(φ))W_{t}

for all ξ ∈ S[a, 1] and φ ∈ S(ℤ^{2}, Ω_{a}), which will then imply that E is F -equivariant in the sense of Proposition 4.5. Of course it is enough to check this identity on the generating set C([a, 1]) ∪ {U_{a}, V_{a}}. Since F acts trivially on C([a, 1]), this amounts to verifying the identities

(ξW_{t})U_{a} = (ξα_{t}(U_{a}))W_{t}, (ξW_{t})V_{a} = (ξα_{t}(V_{a}))W_{t}, and (ξW_{t})f = (ξf)W_{t},

for all f ∈ C([a, 1]). The last equation is easy, so we only have to consider the first two equations. For F = ℤ_{4}, these equations were used (but not explicitly checked) in Section 3 of [72]. The case F = ℤ_{2} follows directly from the case F = ℤ_{4}, since the operator in the quadratic case is the square of the operator in the quartic case. We do the computations for the case F = ℤ_{6} below. The case F = ℤ_{3} will then follow from this in the same way as the case ℤ_{2} follows from the case ℤ_{4}.

The computations in the case that t is the generator of ℤ_{6} are basically the same as the ones given by Walters in Section 3 of [75]. By analogy with Lemma 4.2, and with g ∈ C([a, 1]) defined by g(θ) = e(−θ/2), we have

α_{t}(U_{a}) = V_{a} and α_{t}(V_{a}) = gU_{a}^{−1}V_{a}.

Using this, we have to check the identities

(ξW_{t})U_{a} = (ξV_{a})W_{t} and (ξW_{t})V_{a} = (ξgU_{a}^{−1}V_{a})W_{t}.

The first of these identities is very easy, and we omit the proof. For the second identity, we compute (4.13)

((ξW_{t})V_{a})(s, θ) = e(s)(ξW_{t})(s, θ) = e(s)\frac{e^{πi/12}}{\sqrt{θ}} ∫_{−∞}^{∞} ξ(x, θ)e(\frac{1}{2}θ^{−1}(2sx − x^{2})) dx.

On the other side, changing the variable x to x + θ at the last step, we have

((ξgU_{a}^{−1}V_{a})W_{t})(s, θ) = e( − \frac{1}{2}θ)\frac{e^{πi/12}}{\sqrt{θ}} ∫_{−∞}^{∞} (ξU_{a}^{−1}V_{a})(x, θ)e(\frac{1}{2}θ^{−1}(2sx − x^{2})) dx

= e( − \frac{1}{2}θ)\frac{e^{πi/12}}{\sqrt{θ}} ∫_{−∞}^{∞} e(x)ξ(x − θ, θ)e(\frac{1}{2}θ^{−1}(2sx − x^{2})) dx

= e( − \frac{1}{2}θ)\frac{e^{πi/12}}{\sqrt{θ}} ∫_{−∞}^{∞} e(x + θ)ξ(x, θ)e(\frac{1}{2}θ^{−1}[2s(x + θ) − (x + θ)^{2}]) dx.

The last expression is equal to ((ξW_{t})V_{a})(s, θ) by (4.13) and the easy to verify identity

e( − \frac{1}{2}θ)e(x + θ)e(\frac{1}{2}θ^{−1}[2s(x + θ) − (x + θ)^{2}]) = e(s)e(\frac{1}{2}θ^{−1}(2sx − x^{2})),

which holds for all s, x and θ. □

Notation 4.7. Let F be one of ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6}, and let E be the F equivariant Amodule of Proposition 4.6. We denote by E^{F} the finitely generated A ⋊ F module constructed from E by the procedure described in Proposition 4.5. We write E_{θ}^{F} and E_{θ} for the evaluations of E^{F} and E at θ ∈ [a, 1].

Thus, with ev_{θ} denoting evaluation at θ (on both A = C^{∗}(ℤ^{2}, Ω_{a}) and A ⋊ F), in K-theory we have

(ev_{θ})_{∗}([E]) = [E_{θ}] ∈ K_{0}(A_{θ}) and (ev_{θ})_{∗}([E^{F} ]) = [E_{θ}^{F} ] ∈ K_{0}(A_{θ} ⋊_{α} F).

We now look at the special value θ = 1, for which we have A_{1}⋊_{α}F ≅ C^{∗}(ℤ^{2}⋊F).

Lemma 4.8. Let F be one of the groups ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6} and let [E_{1}^{F}] ∈ K_{0}(C^{∗}(ℤ^{2} ⋊ F)) be as in Notation 4.7 for θ = 1. Then [E_{1}^{F} ] is mapped onto a generator of the group K̃_{0}(C^{∗}(ℤ^{2})) ≅ ℤ under the projection

pr ∘ res_{ℤ^{2}⋊F}^{ℤ^{2}} : K_{0}(C^{∗}(ℤ^{2} ⋊ F)) → K̃_{0}(C^{∗}(ℤ^{2}))

which appears in the sequence of Theorem 3.5.

Proof. Since res_{ℤ^{2}⋊F}^{ℤ^{2}} ([E_{1}^{F}]) = [E_{1}], it suffices to show that ([1], [E_{1}]) is a basis for K_{0}(C^{∗}(ℤ^{2})). By Remark 2.3 (applied to the trivial group H = {1}) it suffices to check that there exists one θ ∈ [a, 1] such that ([1], [E_{θ}]) is a basis for K_{0}(A_{θ}). Choose θ ∈ [a, 1] ∖ ℚ and let τ: A_{θ} → ℝ be the canonical tracial state on A_{θ}. In Theorem 1.4 of [63] it is shown that τ_{∗}([E_{θ}]) = θ. It follows from the Appendix in [55] that τ_{∗} is an isomorphism from K_{0}(A_{θ}) to ℤ + θℤ, so the result follows. □

We are now ready to give an explicit basis for K_{0}(A_{θ} ⋊ F) for all θ ∈ (0, 1] and for F = ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6}.

Theorem 4.9. Let θ ∈ (0, 1]. Let u_{θ}, v_{θ,} and t_{θ} be the generators of A_{θ} ⋊_{α} F = C^{∗}(ℤ^{2}⋊F, ω̃_{θ}) of Lemma 4.2. Choose a with 0 < a ≤ θ, and let [E_{θ}^{F} ] ∈ K_{0}(A_{θ}⋊_{α}F) be as in Notation 4.7. For n = 2, 3, 4, 6 let τ_{n,θ} denote the canonical tracial state on A_{θ} ⋊_{α} ℤ_{n}. Then:

(1) If F = ℤ_{2}, define

p^{θ} = \frac{1}{2}(1 + t_{θ}), r^{θ} = \frac{1}{2}(1 − e(\frac{1}{2}θ)u_{θ}v_{θ}t_{θ}),
q_{0}^{θ} = \frac{1}{2}(1 − u_{θ}t_{θ}),  and  q_{1}^{θ} = \frac{1}{2}(1 − v_{θ}t_{θ}).

Then the classes

[1], [p^{θ}], [q_{0}^{θ}], [q_{1}^{θ}], [r^{θ}], [E_{θ}^{ℤ_{2}}]

form a basis for K_{0}(A_{θ} ⋊_{α} ℤ_{2}) ≅ ℤ^{6}. Moreover, (τ_{2,θ})_{∗} takes the following values (in this order) on these classes:

1, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{θ}{2}.

(2) If F = ℤ_{3}, set ζ = \frac{1}{2}( − 1 + i\sqrt{3}), and define
p_{0}^{θ} = \frac{1}{3}(1 + t_{θ} + t_{θ}^{2}), p_{1}^{θ} = \frac{1}{3}(1 + ζt_{θ} + (ζt_{θ})^{2}),
q_{0}^{θ} = \frac{1}{3}(1 + e(\frac{1}{6}[2 + θ])u_{θ}t_{θ} + e(\frac{1}{3}[2 + θ])(u_{θ}t_{θ})^{2}),
q_{1}^{θ} = \frac{1}{3}(1 + e(\frac{1}{6}[2 + θ])ζu_{θ}t_{θ} + e(\frac{1}{3}[2 + θ])(ζu_{θ}t_{θ})^{2}),
r_{0}^{θ} = \frac{1}{3}(1 + u_{θ}^{2}t_{θ} + (u_{θ}^{2}t_{θ})^{2}),  and  r_{1}^{θ} = \frac{1}{3}(1 + ζu_{θ}^{2}t_{θ} + (ζu_{θ}^{2}t_{θ})^{2}).

Then the classes

[1], [p_{0}^{θ}], [p_{1}^{θ}], [q_{0}^{θ}], [q_{1}^{θ}], [r_{0}^{θ}], [r_{1}^{θ}], [E_{θ}^{ℤ_{3}}]

form a basis for K_{0}(A_{θ} ⋊_{α} ℤ_{3}) ≅ ℤ^{8}. Moreover, (τ_{3,θ})_{∗} takes the following values on these classes:

1, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{θ}{3}.

(3) If F = ℤ^{4}, define

p_{0}^{θ} = \frac{1}{4}(1 + t_{θ} + t_{θ}^{2} + t_{θ}^{3}), p_{1}^{θ} = \frac{1}{4}(1 + it_{θ} − t_{θ}^{2} − it_{θ}^{3}),
p_{2}^{θ} = \frac{1}{4}(1 − t_{θ} + t_{θ}^{2} − t_{θ}^{3}), r^{θ} = \frac{1}{2}(1 − u_{θ}t_{θ}^{2}),
q_{0}^{θ} = \frac{1}{4}(1 + ie(\frac{1}{4}θ)u_{θ}t_{θ} − [e(\frac{1}{4}θ)u_{θ}t_{θ}]^{2} − i[e(\frac{1}{4}θ)u_{θ}t_{θ}]^{3}),
q_{1}^{θ} = \frac{1}{4}(1 − e(\frac{1}{4}θ)u_{θ}t_{θ} + [e(\frac{1}{4}θ)u_{θ}t_{θ}]^{2} − [e(\frac{1}{4}θ)u_{θ}t_{θ}]^{3}),

and

q_{2}^{θ} = \frac{1}{4}(1 − ie(\frac{1}{4}θ)u_{θ}t_{θ} − [e(\frac{1}{4}θ)u_{θ}t_{θ})^{2} + i[e(\frac{1}{4}θ)u_{θ}t_{θ}]^{3}).

Then the classes

[1], [p_{0}^{θ}], [p_{1}^{θ}], [p_{2}^{θ}], [q_{0}^{θ}], [q_{1}^{θ}], [q_{2}^{θ}], [r^{θ}], [E_{θ}^{ℤ_{4}}]

form a basis for K_{0}(A_{θ} ⋊_{α} ℤ_{4}) ≅ ℤ^{9}. Moreover, (τ_{4,θ})_{∗} takes the following values on these classes:

1, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{2}, \frac{θ}{4}.

(4) If F = ℤ_{6}, set ζ = \frac{1}{2}(1 + i\sqrt{3}), define

p_{j}^{θ} = \frac{1}{6}(1 + (ζ^{j}t_{θ}) + (ζ^{j}t_{θ})^{2} + (ζ^{j}t_{θ})^{3} + (ζ^{j}t_{θ})^{4} + (ζ^{j}t_{θ})^{5})

for 0 ≤ j ≤ 4, and define

q_{0}^{θ} = \frac{1}{3}(1 + e(\frac{1}{6}(2 + θ))u_{θ}t_{θ}^{2} + [e(\frac{1}{6}(2 + θ))u_{θ}t_{θ}^{2}]^{2}),
q_{1}^{θ} = \frac{1}{3}(1 + ζ^{2}e(\frac{1}{6}(2 + θ))u_{θ}t_{θ}^{2} + [ζ^{2}e(\frac{1}{6}(2 + θ))u_{θ}t_{θ}^{2}]^{2}),

and

r^{θ} = \frac{1}{2}(1 − u_{θ}t_{θ}^{3}).

Then the classes

[1], [p_{0}^{θ}], [p_{1}^{θ}], [p_{2}^{θ}], [p_{3}^{θ}], [p_{4}^{θ}], [q_{0}^{θ}], [q_{1}^{θ}], [r^{θ}], [E_{θ}^{ℤ_{6}}]

form a basis for K_{0}(A_{θ} ⋊_{α} ℤ_{6}) ≅ ℤ^{10}. Moreover, (τ_{6,θ})_{∗} takes the following values on these classes:

1, \frac{1}{6}, \frac{1}{6}, \frac{1}{6}, \frac{1}{6}, \frac{1}{6}, \frac{1}{3}, \frac{1}{3}, \frac{1}{2}, \frac{θ}{6}.

Proof. In each case, we define projections in C^{∗}(ℤ^{2} ⋊_{α} F, Ω̃_{a}) as follows. In the formulas in Example 3.7 (excluding S, which we treat below), substitute for t, ut, etc. the unitaries given in Lemma 4.4 whose images in the fiber at 1 are t, ut, etc. Thus, for example, for F = ℤ_{6} we replace ut^{2} by φ_{2}U_{a}T_{a}^{2}, and in place of the projection q_{0} we obtain Q_{0} = \frac{1}{3}(1 + φ_{2}U_{a}T_{a}^{2} + (φ_{2}U_{a}T_{a}^{2})^{2}). In each case, by Lemma 4.4, the evaluation at θ ∈ [0, 1] of the projection obtained this way is the corresponding projection in the statement of the theorem. For example, for F = ℤ_{6} we get ev_{θ}(Q_{0}) = q_{0}^{θ}. To get our replacement for S, choose a projection in some matrix algebra over C^{∗}(ℤ^{2} ⋊ F, Ω̃_{a}) corresponding to the module E^{F} as in Notation 4.7. Thus, for each choice of F, we have a collection P of projections in C^{∗}(ℤ^{2} ⋊ F, Ω̃_{a}) or a matrix algebra over this algebra.

Evaluate at θ = 1. The classes of the projections ev_{1}(p) for p ∈ P are exactly the classes listed in Example 3.7, except with S replaced by [E_{θ}^{F} ]. By Example 3.7 and Lemma 4.8, these classes form a basis for K_{0}(C^{∗}(ℤ^{2} ⋊ F)). By Remark 2.3, for any θ ∈ [a, 1], the classes of the projections ev_{θ}(p) for p ∈ P form a basis for K_{0}(A_{θ} ⋊_{α} F).

The computation of the trace is clear from the description for all classes except the classes [E_{θ}^{F} ]. The values of the traces of the classes [E_{θ}^{F} ] are computed in Proposition 3.3 of [72] for F = ℤ_{4} and in Theorem 2 of [75] for F = ℤ_{3}, ℤ_{6}. The case F = ℤ_{2} is done similarly. □

5. The tracial Rokhlin property

In this section, we prove that for θ ∈ ℝ ∖ ℚ the actions of ℤ_{2}, ℤ_{3}, ℤ_{4}, and ℤ_{6} on A_{θ}, given by (0.1) for the groups generated as in (0.2), have the tracial Rokhlin property, and that for every nondegenerate skew-symmetric matrix Θ, the flip action of ℤ_{2} on A_{Θ} has the tracial Rokhlin property. We do this by showing that, under suitable conditions, the tracial Rokhlin property is equivalent to outerness of the corresponding action on the type II_{1} factor obtained as the weak operator closure of A in the Gelfand-Naimark-Segal representation associated with the tracial state. A preliminary result, stated entirely in terms of C*-algebras, gives a criterion for the tracial Rokhlin property in terms of trace norms. For use elsewhere, we state and prove this result for C*-algebras with an arbitrary simplex of tracial states.

We will make extensive use of the L^{2}-norm (or seminorm) associated with a tracial state τ of a C*-algebra A, given by ∥a∥_{2,τ} = τ(a^{∗}a)^{1/2}. See the discussion before Lemma V.2.20 of [70] for more on this seminorm in the von Neumann algebra context. All the properties we need are immediate from its identification with the seminorm in which one completes A to obtain the Hilbert space H_{τ} for the Gelfand-Naimark-Segal representation associated with τ, and from the relation τ(ba) = τ(ab). In particular, we always have ∥abc∥_{2,τ} ≤ ∥a∥ · ∥b∥_{2,τ} · ∥c∥.

Since the C*-algebras in this section will almost all be finite, we recall the tracial Rokhlin property for actions of finite groups on finite infinite dimensional simple unital C*-algebras. The following result is Lemma 1.16 of [59].

Proposition 5.1. Let A be a finite infinite dimensional simple unital C*-algebra, and let α: G → Aut(A) be an action of a finite group G on A. Then α has the tracial Rokhlin property if and only if for every finite set S ⊆ A, every ε > 0, and every nonzero positive element x ∈ A, there are mutually orthogonal projections e_{g} ∈ A for g ∈ G such that:

(1) ∥α_{g}(e_{h}) − e_{gh}∥ < ε for all g, h ∈ G.

(2) ∥e_{g}a − ae_{g}∥ < ε for all g ∈ G and all a ∈ S.

(3) With e = ∑_{g∈G} e_{g}, the projection 1 − e is Murray-von Neumann equivalent to a projection in the hereditary subalgebra of A generated by x.

We begin with a simple reformulation of the tracial Rokhlin property which is valid in the presence of good comparison properties for projections. Recall that, if A is a unital C*-algebra, then we say that the order on projections over A is determined by traces if Blackadar’s Second Fundamental Comparability Question (1.3.1 in [4]) holds for all matrix algebras over A. That is, whenever n ∈ ℕ and p, q ∈ M_{n}(A) are projections such that τ(p) < τ(q) for all tracial states τ on A, then p ≾ q.

Lemma 5.2. Let A be a finite infinite dimensional simple separable unital C*- algebra with Property (SP) (every nonzero hereditary subalgebra contains a nonzero projection) and such that the order on projections over A is determined by traces. Let α: G → Aut(A) be an action of a finite group G on A. Then α has the tracial Rokhlin property if and only if for every finite set S ⊆ A and every ε > 0, there exist orthogonal projections e_{g} ∈ A for g ∈ G such that:

(1) ∥α_{g}(e_{h}) − e_{gh}∥ < ε for all g, h ∈ G.

(2) ∥e_{g}a − ae_{g}∥ < ε for all g ∈ G and all a ∈ S.
(3) With e = ∑_{g∈G} e_{g}, we have τ(1 − e) < ε for all τ ∈ T (A).

Proof. Suppose that the condition of Proposition 5.1 holds, and let S and ε be given. Choose n ∈ ℕ with n > 1/ε. By Lemma 1.10 of [59], there exist n nonzero mutually orthogonal projections in A. Let x be one of them. Then τ(x) < ε for all τ ∈ T (A). Apply Proposition 5.1 with this x and with S and ε as given. Conversely, assume the condition of the lemma, and let S, ε, and x be given. Using Property (SP), choose a nonzero projection q ∈ \overline{xAx}, and apply the condition of the lemma with ε replaced by min (ε, inf_{τ∈T (A)} τ(q)) . The assumption that the order on projections over A is determined by traces implies that 1 − e ≾ q, giving (3) of Proposition 5.1. □

Theorem 5.3. Let A be an infinite dimensional simple separable unital C*-algebra with tracial rank zero. Let α: G → Aut(A) be an action of a finite group G on A. Then α has the tracial Rokhlin property if and only if for every finite set S ⊆ A and every ε > 0, there exist orthogonal projections e_{g} ∈ A for g ∈ G such that:

(1) ∥α_{g}(e_{h}) − e_{gh}∥_{2,τ} < ε for all g, h ∈ G and all τ ∈ T (A).

(2) ∥[e_{g}, a]∥_{2,τ} < ε for all g ∈ G, all a ∈ S, and all τ ∈ T (A).
(3) ∑_{g∈G} e_{g} = 1.

The equivalence is also valid if one substitutes for Condition (3) the following condition:

(3′) τ (1 − ∑_{g∈G} e_{g}) < ε for all τ ∈ T (A).

Proof. We first prove that if α has the tracial Rokhlin property, then the version of the condition of the theorem using Conditions (1), (2), and (3′) holds. Let S ⊆ A be finite and let ε > 0. Since A has tracial rank zero, Corollary 5.7 and Theorems 5.8 and 6.8 of [40] imply that the order on projections over A is determined by traces. Therefore we may apply Lemma 5.2 with S and ε as given. Since ∥a∥_{2,τ} ≤ ∥a∥ for all τ ∈ T (A), it is easy to show that the resulting projections satisfy (1), (2), and (3′).

Next, assume that the version with Conditions (1), (2), and (3′) holds. Let S ⊆ A be finite and let ε > 0. We may assume that ∥a∥ ≤ 1 for all a ∈ S. Apply the hypothesis with S as given and with min (\frac{1}{3}ε, \frac{1}{9}ε^{2}) in place of ε. Call the resulting projections p_{g} for g ∈ G. Set p = ∑_{g∈G} p_{g}, set e_{1} = p_{1} + 1 − p, and set e_{g} = p_{g} for g ∈ G ∖ \{1\}. Since ∥e_{1} − p_{1}∥_{2,τ} = τ(1 − p)^{1/2} < \frac{1}{3}ε for all τ ∈ T (A), it is easy to show that the e_{g} satisfy (1), (2), and (3).

Now we prove that if (1), (2), and (3) hold, then α has the tracial Rokhlin property. We follow the proof of Theorem 2.14 of [51] (but note that the finite set there is called F). We describe the choices and constructions carefully, but omit many of the details in the verification of the estimates. We verify the condition of Lemma 5.2. So let S ⊆ A be a finite set, and let ε > 0. Without loss of generality ε < 1.

Choose ε_{0} > 0 with

ε_{0} < \frac{ε}{3card(G)}

and so small that whenever p_{g}, for g ∈ G, are projections in a C*-algebra B which satisfy ∥p_{g}p_{h}∥ < 4ε_{0} for g ≠ h, then there are orthogonal projections e_{g} ∈ B, for g ∈ G, such that e_{1} = p_{1} and ∥e_{g} − p_{g}∥ < \frac{1}{3}ε for g ∈ G.

Apply Lemma 2.8 of [51] with ε_{0} in place of ε and with n = card(G) − 1, and let ε_{1} > 0 be the resulting value of δ. We also require ε_{1} ≤ ε_{0}. Then set

ε_{2} = min (1, \frac{ε_{1}^{2}}{16}, \frac{ε_{1}}{8} , \frac{ε}{18}) .

Set T = ⋃_{g∈G} α_{g}(S). Apply Lemma 2.13 of [51] with ε_{2} in place of ε, with T in place of F, and with G in place of S. We obtain projections q, q_{0} ∈ A, unital finite dimensional subalgebras E ⊆ qAq and E_{0} ⊆ q_{0}Aq_{0}, and automorphisms φ_{g} ∈ Aut(A) for g ∈ G, such that:

(1) φ_{1} = id_{A} and ∥φ_{g} − α_{g}∥ < ε_{2} for all g ∈ G.

(2) For every g ∈ G and x ∈ E, we have q_{0}φ_{g}(x) = φ_{g}(x)q_{0} and q_{0}φ_{g}(x)q_{0} ∈ E_{0}.

(3) For every a ∈ T, we have ∥qa − aq∥ < ε_{2} and dist(qaq, E) < ε_{2}.

(4) τ(1 − q), τ(1 − q_{0}) < ε_{2} for all τ ∈ T (A).

Apply Lemma 2.12 of [51] with ε_{2} in place of ε and E_{0} + ℂ(1 − q_{0}) in place of E, obtaining δ > 0. Also require δ ≤ ε_{2}.

Apply the hypothesis with δ in place of ε, and with a system of matrix units for E_{0} + ℂ(1 − q_{0}) in place of S, getting projections p_{g} for g ∈ G. Let B_{0} = A ∩ [E_{0} + ℂ(1 − q_{0})]′, the subalgebra of A consisting of all elements which commute with everything in E_{0} +ℂ(1−q_{0}). Apply the choice of δ using Lemma 2.12 of [51] to p_{1}, obtaining a projection f ∈ B_{0} which satisfies ∥f − p_{1}∥_{2,τ} < ε_{2} for all τ ∈ T (A). Since q_{0} is in the center of E_{0} + ℂ(1 − q_{0}), the element f_{1} = q_{0}f is also a projection in B_{0}. For g ∈ G, since q_{0}φ_{g^{−1}}(E)q_{0} ⊆ E_{0}, it follows that f_{1} commutes with all elements of q_{0}φ_{g^{−1}}(E)q_{0} and hence with all elements of φ_{g^{−1}}(E). Therefore f_{g} = φ_{g}(f_{1}) commutes with all elements of E, including q. So f_{g} also commutes with 1 − q.

Following the analogous estimates in the proof of Theorem 2.14 of [51], we get ∥f_{1} − p_{1}∥_{2,τ} < \frac{1}{4}ε_{1} + ε_{2} for τ ∈ T (A) and ∥f_{g} − α_{g}(f_{1})∥ < ε_{2} for g ∈ G. We then get (with a slight difference: the sum in the estimate in [51] is not needed here),

∥f_{g} − p_{g}∥_{2,τ} < ε_{2} + (\frac{1}{4}ε_{1} + ε_{2}) + ε_{2}

for τ ∈ T (A). Continuing as there, for g ∈ G ∖ {1} we get

∥f_{1}f_{g}∥_{2,τ} = ∥f_{1}f_{g} − p_{1}p_{g}∥_{2,τ} < \frac{1}{2}ε_{1} + 4ε_{2} ≤ ε_{1}.

We saw that f_{g} ∈ B = A ∩ [E + ℂ(1 − q)]′ for g ∈ G. This algebra has real rank zero because E + ℂ(1 − q) is finite dimensional. Therefore Lemma 2.8 of [51], applied to B with {τ|_{B}: τ ∈ T (A)} in place of T, and the choice of ε_{1}, provide a projection r ∈ A ∩ [E + ℂ(1 − q)]′ such that r ≤ f_{1}, such that ∥rf_{g}∥ < ε_{0} for g ∈ G, and such that τ(r) > τ(f_{1}) − ε_{0} for all τ ∈ T (A).

Now use Corollary 2.4 of [51] to find a projection e_{1} ∈ A ∩ [E + ℂ(1 − q)]′ such that e_{1} ≤ q, such that ∥re_{1}−e_{1}∥ < ε_{0}, and such that [e_{1}] ≥ [1]−([1]−[q])−([1]−[r]) in K_{0}(A ∩ [E + ℂ(1 − q)]′). The last inequality implies that

τ(e_{1}) ≥ τ(r) − τ(1 − q) > τ(f_{1}) − ε_{0} − ε_{2} ≥ τ(f_{1}) − 2ε_{0}

for all τ ∈ T (A).

We now show that the α_{g}(e_{1}), for g ∈ G, are approximately orthogonal. It suffices to estimate ∥e_{1}α_{g}(e_{1})∥ for g ∈ G. We first use φ_{g}(r) ≤ φ_{g}(f_{1}) = f_{g} to get ∥rφ_{g}(r)∥ ≤ ∥rf_{g}∥ < ε_{0}. Then, following the proof of Theorem 2.14 of [51], for g ∈ G we get ∥e_{1}φ_{g}(e_{1})∥ < 3ε_{0} and

∥e_{1}α_{g}(e_{1})∥ ≤ ∥α_{g} − φ_{g}∥ + ∥e_{1}φ_{g}(e_{1})∥ < 4ε_{0}.

Now use the choice of ε_{0} to find orthogonal projections e_{g} ∈ A∩[E+ℂ(1−q)]′ for g ∈ G∖\{1\}, all orthogonal to e_{1}, such that ∥e_{g} −α_{g}(e_{1})∥ < \frac{1}{3}ε for g ∈ G∖\{1\}. Then the e_{g}, for g ∈ G, satisfy Conditions (1) through (3) of Lemma 5.2, by estimates essentially the same as in the last part of the proof of Theorem 2.14 of [51]. □

Notation 5.4. If A is a simple C*-algebra with unique tracial state τ, and α ∈ Aut(A), we write α′′ for the automorphism of π_{τ} (A)′′ determined by α.

Theorem 5.5. Let A be a simple separable unital C*-algebra with tracial rank zero, and suppose that A has a unique tracial state τ. Let π_{τ}: A → B(H_{τ}) be the Gelfand-Naimark-Segal representation associated with τ. Let G be a finite group, and let α: G → Aut(A) be an action of G on A. Then α has the tracial Rokhlin property if and only if α′′_{g} is an outer automorphism of π_{τ} (A)′′ for every g ∈ G∖{1}.

Proof. Assume that α′′_{g} is outer for every g ∈ G ∖ {1}. We verify the hypotheses of Theorem 5.3. Thus let ε > 0 and let S ⊆ A be a finite subset. Without loss of generality ∥a∥ ≤ 1 for all a ∈ S. Set n = card(G), and set ε_{0} = (4n + 1)^{−1}ε. Choose δ > 0 as in Lemma 2.9 of [51] with n as given and with ε_{0} in place of ε.

We regard A as a subalgebra of N = π_{τ} (A)′′, and we let τ also denote the extension of τ to N. The algebra N is hyperfinite by Lemma 2.16 of [51].

We apply Lemma 5.2.1 of [35]. The term “equivariant s.m.u.” [system of matrix units] is defined in Section 1.5 of [35], and N(ψ) is defined in Section 1.2 of [35]. (Also note that pages 44 and 45 of [35] are switched.) We take the set X of Lemma 5.2.1 of [35] to be G with the left translation action. The lemma provides, ignoring the off diagonal matrix units, projections p_{g} ∈ π_{τ} (A)′′ such that α′′_{g}(p_{h}) = p_{gh} for g, h ∈ G, and such that ∥[p_{g}, a]∥_{2,τ} < δ for g ∈ G and a ∈ S.

For g ∈ G use Lemma 2.15 of [51] to find a projection q_{g} ∈ A such that ∥q_{g} − p_{g}∥_{2,τ} < min (\frac{1}{2}δ, ε_{0}) . Then ∥q_{g}q_{h}∥_{2,τ} < δ for g ≠ h, so the choice of δ using Lemma 2.9 of [51] provides mutually orthogonal projections e_{g} ∈ A for g ∈ G ∖ {1} such that ∥e_{g} − q_{g}∥_{2,τ} < ε_{0}, and therefore ∥e_{g} − p_{g}∥_{2,τ} < 2ε_{0}. Set e_{1} = 1 − ∑_{g∈G∖{1}} e_{g}. Since ∑_{g∈G} p_{g} = 1, we get

∥e_{1} − p_{1}∥_{2,τ} ≤ ∑_{g∈G∖{1}} ∥e_{g} − p_{g}∥_{2,τ} < 2(n − 1)ε_{0}.

So ∥e_{g} − p_{g}∥_{2,τ} < 2nε_{0} for all g ∈ G.

For g, h ∈ G, using α′′_{g}(p_{h}) = p_{gh} we now get

∥α_{g}(e_{h}) − e_{gh}∥_{2,τ} ≤ ∥α′′_{g}(e_{h} − p_{h})∥_{2,τ} + ∥p_{gh} − e_{gh}∥_{2,τ} < 2nε_{0} + 2nε_{0} < ε.

Also, if a ∈ S and g ∈ G then

∥[a, e_{g}]∥_{2,τ} ≤ 2∥e_{g} − p_{g}∥_{2,τ} · ∥a∥ + ∥[a, p_{g}]∥_{2,τ} < 2 · 2nε_{0} + ε_{0} ≤ ε.

This completes the proof that outerness in the trace representation implies the tracial Rokhlin property.

The proof of the converse is similar to the proof of Lemma 1.5 of [59]. Set n = card(G) as before, and set

ε = min (\frac{1}{2}, \frac{1}{4\sqrt{2n}}) .

Let g_{0} ∈ G∖{1}, and let u ∈ π_{τ} (A)′′ be a unitary; we show that α′′_{g_{0}} ≠ Ad(u). Let ξ_{τ} be the standard cyclic vector for π_{τ} , so that ∥x∥_{2,τ} = ∥xξ_{τ} ∥ for all x ∈ π_{τ} (A)′′. The Kaplansky Density Theorem implies that the unit ball of π_{τ} (A) is strong operator dense in the unit ball of π_{τ} (A)′′. Therefore there is a ∈ A with ∥a∥ ≤ 1 such that ∥u − a∥_{2,τ} < ε. Choose a nonzero projection f ∈ A with τ(f) < ε, and choose mutually orthogonal projections e_{g} ∈ A for g ∈ G such that ∥e_{g}a − ae_{g}∥ < ε for g ∈ G, such that ∥α_{g}(e_{h}) − e_{gh}∥ < ε for g, h ∈ G, and such that 1 − ∑_{g∈G} e_{g} ≾ f. Since τ is the unique tracial state, it is G-invariant, so that

τ(e_{1}) = \frac{1}{n} ∑_{g∈G} τ(e_{g}) ≥ \frac{1}{n}(1 − τ(f)) ≥ \frac{1}{2n}.

Now

∥e_{1} − e_{g_{0}}∥_{2,τ} ≤ 2∥u − a∥_{2,τ} + ∥ae_{1} − e_{1}a∥ + ∥ue_{1}u^{∗} − α_{g_{0}}(e_{1})∥ + ∥α_{g_{0}}(e_{1}) − e_{g_{0}}∥
< 2ε + ε + ∥ue_{1}u^{∗} − α_{g_{0}}(e_{1})∥ + ε.

On the other hand, using orthogonality,

∥e_{1} − e_{g_{0}}∥_{2,τ} ≥ ∥e_{1}∥_{2,τ} ≥ \frac{1}{\sqrt{2n}}.

Therefore

∥ue_{1}u^{∗} − α_{g_{0}}(e_{1})∥ > \frac{1}{\sqrt{2n}} − 4ε ≥ 0.

Thus ue_{1}u^{∗} ≠ α_{g_{0}}(e_{1}), and α′′_{g_{0}} ≠ Ad(u). □

In case the group is ℤ, Theorem 2.18 of [51] gives two more equivalent conditions for an action α: G → Aut(A) on a simple unital C*-algebra with tracial rank zero and unique tracial state to have the tracial Rokhlin property, namely that the crossed product A⋊_{α} G have real rank zero, and that it have a unique tracial state. For finite groups, the tracial Rokhlin property implies these conditions, but neither condition implies the tracial Rokhlin property.

The tracial Rokhlin property implies that the crossed product has real rank zero under these hypotheses, because it implies that the crossed product has tracial rank zero (Theorem 2.6 of [59]). The converse is easily seen to be false, by considering the trivial action of any finite group on any simple unital C*-algebra with tracial rank zero and a unique tracial state. Example 2.9 of [61] shows that it does not help to require that the action be outer.

We now consider the condition that A ⋊_{α} G have a unique tracial state. One can give a direct argument that this follows from the tracial Rokhlin property. The line of reasoning we give here uses some machinery and an additional useful lemma. For this lemma, we do not require the algebra to be finite, so the tracial Rokhlin property is as in Definition 1.2 of [59].

Lemma 5.6. Let A be an infinite dimensional simple unital C*-algebra, and let α: G → Aut(A) be an action of a finite group G on A which has the tracial Rokhlin property. Let H be a subgroup of G. Then α|_{H} has the tracial Rokhlin property.

Proof. Set m = card(G)/card(H). Choose a set S of right coset representatives for H in G, so that card(S) = m and every element of G can be written uniquely as hs with h ∈ H and s ∈ S.

Let S ⊆ A be finite, let ε > 0, and let x ∈ A be a positive element with ∥x∥ = 1. Set ε_{0} = ε/m. The tracial Rokhlin property for α provides mutually orthogonal projections p_{g} ∈ A for g ∈ G such that:

(1) ∥α_{g}(p_{h}) − p_{gh}∥ < ε_{0} for all g, h ∈ G.

(2) ∥p_{g}a − ap_{g}∥ < ε_{0} for all g ∈ G and all a ∈ S.

(3) With p = ∑_{g∈G} p_{g}, the projection 1−p is Murray-von Neumann equivalent to a projection in the hereditary subalgebra of A generated by x.

(4) With p as in (3), we have ∥pxp∥ > 1 − ε. For h ∈ H define e_{h} = ∑_{s∈S} p_{hs}. These are clearly mutually orthogonal projections. We verify the analogs of Conditions (1) through (4). For h, k ∈ H, we have

∥α_{k}(e_{h}) − e_{kh}∥ ≤ ∑_{s∈S} ∥α_{k}(p_{hs}) − p_{khs}∥ < mε_{0} = ε.

For h ∈ H and a ∈ S, we have

∥e_{h}a − ae_{h}∥ ≤ ∑_{s∈S} ∥p_{hs}a − ap_{hs}∥ < mε_{0} = ε.

The projection e = ∑_{h∈H} e_{h} is equal to p, so that 1 − e = 1 − p is Murray-von Neumann equivalent to a projection in the hereditary subalgebra of A generated by x. Finally, ∥exe∥ = ∥pxp∥ > 1 − ε. □

Proposition 5.7. Let A be an infinite dimensional simple unital C*-algebra, and let α: G → Aut(A) be an action of a finite group G on A which has the tracial Rokhlin property. Let r: T (A ⋊_{α} G) → T (A) be the restriction map between the tracial state spaces. Then r is an affine homeomorphism from T (A ⋊_{α} G) to the subspace T (A)^{G} of G-invariant tracial states on A.

Proof. It is easy to see that the image of r is exactly T (A)^{G}, and that r is continuous and affine. It therefore suffices to show that r is injective, that is, that a tracial state τ on A ⋊_{α} G is uniquely determined by its values on A. We prove this by proving that if a ∈ A, if g ∈ G ∖ {1}, and if u_{g} ∈ A ⋊_{α} G is the corresponding unitary, then τ(au_{g}) = 0.

Let H be the subgroup of G generated by g, and let β = α|_{H}. Set σ = τ|_{A⋊_{β}H}. Then β has the tracial Rokhlin property by Lemma 5.6. Therefore β̂ is tracially approximately representable, by Theorem 3.12(1) of [59]. Choose μ ∈ Ĥ such that μ(g) ≠ 1. It follows from Proposition 6.1 of [59] that σ ∘ β̂_{μ} = σ. Since β̂_{μ}(au_{g}) = \overline{μ(g)}au_{g}, this implies that σ(au_{g}) = 0. So τ(au_{g}) = 0. □

In particular, if A has a unique tracial state and α: G → Aut(A) has the tracial Rokhlin property, then A ⋊_{α} G has a unique tracial state.

Again, the converse is false.

Example 5.8. First take A = M_{2} and G = (ℤ_{2})^{2}. Let the two generators act by conjugation by the unitaries ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) and ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) . Because the matrices commute up to a scalar, one gets a well defined action α: G → Aut(M_{2}), such that α_{g} is inner for every g ∈ G. It is well known (see, for example, Example 4.2.3 of [56]) that A ⋊_{α} G ≅ M_{4}, which has a unique tracial state.

This example is finite dimensional, and the tracial Rokhlin property was only defined for infinite dimensional C*-algebras. To get an infinite dimensional example, tensor the above example with, say, the trivial action of G on the 2^{∞} UHF algebra.

The following lemma is well known, and is given in this form as Lemma 3.1 of [51].

Lemma 5.9. Let A be a separable unital C*-algebra with a faithful tracial state τ. Let (y_{i})_{i∈I} be a family of unitaries such that τ(y_{i}^{∗}y_{j}) = 0 for i ≠ j and whose linear span is dense in A. Let π: A → B(H) be the Gelfand-Naimark-Segal representation associated with τ. We identify A with its image in π(A)′′. Then every a ∈ π(A)′′ has a unique representation as a = ∑_{i∈I} λ_{i}y_{i}, with convergence in ∥·∥_{2,τ} and whose coefficients satisfy ∑_{i∈I} |λ_{i}|^{2} = ∥a∥_{2,τ}^{2} . If a is unitary then ∑_{i∈I} |λ_{i}|^{2} = 1.

Recall that a real skew symmetric d × d matrix Θ is nondegenerate if whenever x ∈ ℤ^{d} satisfies exp(2πi⟨x, Θy⟩) = 1 for all y ∈ ℤ^{d}, then x = 0. This definition is essentially from Section 1.1 of [69]. It is well known that the noncommutative torus A_{Θ} (defined after Corollary 1.13) is simple if and only if Θ is nondegenerate; the main part is Theorem 3.7 of [69], and the complete statement is Theorem 1.9 of [60].

Lemma 5.10. Let Θ be a nondegenerate real skew symmetric d × d matrix. Let 1 ≤ k ≤ d, and let α ∈ Aut(A_{Θ}) be an automorphism such that α(u_{k}) = ρu_{1}^{m_{1}}u_{2}^{m_{2}} ⋯ u_{d}^{m_{d}} for some ρ ∈ ℂ with |ρ| = 1, and for some m ∈ ℤ^{d} not equal to the kth standard basis vector δ_{k}. Then α′′ (as in Notation 5.4) is outer.

Proof. Without loss of generality k = 1.

Let τ be the unique tracial state on A_{Θ}, and identify A_{Θ} with its image in π_{τ} (A_{Θ})′′.

Suppose α′′ is inner, so α′′ = Ad(w) for some unitary w ∈ π_{τ} (A_{Θ})′′. We apply Lemma 5.9 with I = ℤ^{d} and y_{n} = u_{1}^{n_{1}}u_{2}^{n_{2}} ⋯ u_{d}^{n_{d}} for n ∈ ℤ^{d}, and write

w^{∗} = ∑_{n∈ℤ^{d}} λ_{n}u_{1}^{n_{1}}u_{2}^{n_{2}} ⋯ u_{d}^{n_{d}}

with convergence in ∥ · ∥_{2,τ} . Using the commutation relations, there are numbers ζ_{n} for n ∈ ℤ^{d}, with |ζ_{n}| = 1, such that

(u_{1}^{n_{1}}u_{2}^{n_{2}} ⋯ u_{d}^{n_{d}})(u_{1}^{m_{1}}u_{2}^{m_{2}} ⋯ u_{d}^{m_{d}}) = ζ_{n}u_{1}^{n_{1}+m_{1}}u_{2}^{n_{2}+m_{2}} ⋯ u_{d}^{n_{d}+m_{d}}.

Then, also with convergence in ∥ · ∥_{2,τ} ,

∑_{n∈ℤ^{d}} λ_{n}u_{1}^{n_{1}+1}u_{2}^{n_{2}} ⋯ u_{d}^{n_{d}} = u_{1}w^{∗} = w^{∗}α(u_{1})

= ∑_{n∈ℤ^{d}} ρζ_{n}λ_{n}u_{1}^{n_{1}+m_{1}}u_{2}^{n_{2}+m_{2}} ⋯ u_{d}^{n_{d}+m_{d}}

= ∑_{n∈ℤ^{d}} ρζ_{n−m+δ_{1}}λ_{n−m+δ_{1}}u_{1}^{n_{1}+1}u_{2}^{n_{2}} ⋯ u_{d}^{n_{d}}.

We have λ_{l} ≠ 0 for some l ∈ ℤ^{d}. Since ρ and the ζ_{n} have absolute value 1, uniqueness of the series representation implies

|λ_{l}| = |λ_{l−m+δ_{1}}| = |λ_{l−2m+2δ_{1}}| = · · · .

Since m ≠ δ_{1}, this contradicts ∑_{n∈ℤ^{d}} |λ_{n}|^{2} < ∞. □

Corollary 5.11. Let F ⊆ SL_{2}(ℤ) be one of the groups ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6}, and let α: F → Aut(A_{θ}) be given by (0.1) for the groups generated as in (0.2). Suppose θ ∉ ℚ. Then α has the tracial Rokhlin property.

Proof. In each case, Lemma 5.10 implies that α′′_{g} is outer for g ∈ G ∖ {1}. Apply Theorem 5.5. □

Let ℤ_{2} act on ℤ^{d} via the flip n ↦ −n. It is immediately clear that for any d × d skew symmetric real matrix Θ, the cocycle ω_{Θ} is invariant under this action. It thus induces an action, also called the flip, of ℤ_{2} on the higher dimensional noncommutative torus A_{Θ}. On the standard generators, this action sends u_{k} to u_{k}^{∗} for 1 ≤ k ≤ d.

Corollary 5.12. Let Θ be a nondegenerate real d×d skew symmetric matrix. Then the flip action on A_{Θ} has the tracial Rokhlin property.

Proof. Lemma 5.10 implies that the flip automorphism is outer. Theorem 3.5 of [60] implies that A_{Θ} is a simple separable unital C*-algebra with tracial rank zero. Apply Theorem 5.5. □

6. The structure of the crossed products

In this section, we prove that for θ ∈ ℝ ∖ ℚ the crossed products A_{θ} ⋊_{α} ℤ_{2}, A_{θ} ⋊_{α} ℤ_{3}, A_{θ} ⋊_{α} ℤ_{4}, and A_{θ} ⋊_{α} ℤ_{6}, by the action (0.1) for the groups generated as in (0.2), are AF algebras, and we determine when two of them are isomorphic. For every nondegenerate Θ, we also prove that the crossed product A_{Θ} ⋊ ℤ_{2} by the flip is an AF algebra. We further prove that all the fixed point algebras are AF. As mentioned in the introduction, this was already known for A_{θ} ⋊_{α}ℤ_{2} for all irrational θ [9], for A_{θ} ⋊_{α} ℤ_{4} for “most” irrational θ [74], and for A_{Θ} ⋊ ℤ_{2} for “most” skew symmetric real matrices Θ [6].

We need to show that the crossed products satisfy the Universal Coefficient Theorem, as in Theorem 1.17 of [67]. A much stronger result than the one formulated in the proposition below has been obtained by Meyer and Nest in Proposition 8.5 of [47]. Since the arguments used in the special case we consider here are much easier, we include the short proof.

Proposition 6.1. Let G be an amenable (or, more generally, an a-T -menable) group which can be embedded as a closed subgroup of some almost connected group L. Then the full and reduced crossed products A ⋊_{α} G and A ⋊_{α,r} G of any type I algebra A by G are KK-equivalent to commutative C*-algebras.

Proof. Let C denote the maximal compact subgroup of L and let T^{∗}(L/C) denote the cotangent space of the quotient L/C (which is a manifold). Then it follows from the work of Kasparov (see Theorem 2 of [36]; Definitions 4.1 and 5.1 and Theorems 5.2 and 5.7 of [37]) that there exist Dirac and dual-Dirac elements

D ∈ KK_{0}^{L}(C_{0}(T^{∗}(L/C)), ℂ) and β ∈ KK_{0}^{L}(ℂ, C_{0}(T^{∗}(L/C)))

such that

D ⊗_{ℂ} β = 1 ∈ KK_{0}^{L}(C_{0}(T^{∗}(L/C)), C_{0}(T^{∗}(L/C)))

and such that γ = β ⊗_{C_{0}(T^{∗}(L/C))} D ∈ KK_{0}^{L}(ℂ, ℂ) is the so-called γ-element of L. The restriction γ_{G} of γ from L to G is the γ-element of G. (For example, see Remark 6.4 of [12].) Since G is a-T -menable, we know from the work of Higson and Kasparov (Theorem 1.1 of [33]) that γ_{G} = 1 ∈ KK_{0}^{G}(ℂ, ℂ). But this implies that ℂ is KK^{G}-equivalent to the commutative algebra C_{0}(T^{∗}(L/C)), on which G acts properly. Now proceed as in the proof of Lemma 5.4 of [15]. □

The algebras in the previous proposition thus satisfy the Universal Coefficient Theorem. In particular, we have:

Corollary 6.2. Let G be an amenable group which can be embedded as a closed subgroup of some almost connected group L. Let ω: G×G → 𝕋 be a Borel 2-cocycle on G. Then C^{∗}(G, ω) satisfies the Universal Coefficient Theorem (Theorem 1.17 of [67]).

Proof. As in the discussion at the beginning of Section 1, there is action α: G → Aut(K) such that C^{∗}(G, ω) ⊗ K ≅ K ⋊_{α} G. Thus Proposition 6.1 implies that C^{∗}(G, ω) is KK-equivalent to a commutative C*-algebra. It therefore satisfies the Universal Coefficient Theorem, by Proposition 7.1 of [67]. □

Theorem 6.3. Let F ⊆ SL_{2}(ℤ) be one of the groups ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6}, and let α: F → A_{θ} be given by (0.1) for the groups generated as in (0.2). Suppose θ ∉ ℚ. Then A_{θ} ⋊_{α} F is an AF algebra.

Proof. We know that A_{θ} is a simple separable unital C*-algebra with tracial rank zero, say by the Elliott-Evans Theorem [25]. (See Theorem 3.5 of [60].) By Corollary 5.11, we may apply Corollary 1.6 and Theorem 2.6 of [59] to conclude that A_{θ}⋊_{α}F is a simple separable unital C*-algebra with tracial rank zero. Corollary 6.2 implies that A_{θ} ⋊_{α} F satisfies the Universal Coefficient Theorem. Therefore Huaxin Lin’s classification theorem (Theorem 5.2 of [41]), in the form presented in Proposition 3.7 of [60], together with the K-theory computation of Corollary 3.8, implies the conclusion. □

The case F = ℤ_{2} is not new [9].

Using the results of Section 4, one can determine exactly which AF algebra the crossed product is. As an example, we show that, for fixed F, the crossed product algebras are isomorphic if and only if the original irrational rotation algebras are isomorphic.

Theorem 6.4. Let F ⊆ SL_{2}(ℤ) be one of the groups ℤ_{2}, ℤ_{3}, ℤ_{4}, ℤ_{6}, and let α: F → A_{θ} be given by (0.1) for the groups generated as in (0.2). Let θ_{1}, θ_{2} ∈ ℝ ∖ ℚ. Then A_{θ_{1}} ⋊_{α} F ≅ A_{θ_{2}} ⋊_{α} F if and only if θ_{1} = ±θ_{2} mod ℤ.

Proof. We give the proof for the case F = ℤ_{2}, using Theorem 4.9(1). The proofs for the other parts are the same, using instead Parts (2), (3), and (4) of Theorem 4.9, and choosing in place of p^{θ} projections with traces \frac{1}{3}, \frac{1}{4}, and \frac{1}{6}.

We clearly have A_{θ} ⋊_{α} ℤ_{2} ≅ A_{θ+n} ⋊_{α} ℤ_{2} for n ∈ ℤ, so without loss of generality θ_{1}, θ_{2} ∈ (0, 1). From Theorem 4.9(1), it is clear that for θ ∈ (0, 1) ∖ ℚ the canonical tracial state τ_{2,θ} on A_{θ} ⋊_{α} ℤ_{2} satisfies

(τ_{2,θ})_{∗}(K_{0}(A_{θ} ⋊_{α} ℤ_{2})) = \frac{1}{2}ℤ + \frac{1}{2}θℤ.

It is well known (and also follows directly from Proposition 5.7 and Corollary 5.11) that τ_{2,θ} is the unique tracial state on A_{θ} ⋊_{α} ℤ_{2}. Accordingly, if θ_{1} ≠ ±θ_{2} mod ℤ, then

2(τ_{2,θ_{1}})_{∗}(K_{0}(A_{θ_{1}} ⋊_{α} ℤ_{2})) = ℤ + θ_{1}ℤ ≠ ℤ + θ_{2}ℤ = 2(τ_{2,θ_{2}})_{∗}(K_{0}(A_{θ_{2}} ⋊_{α} ℤ_{2})) whence A_{θ_{1}} ⋊_{α} ℤ_{2} ≅ A̸_{θ_{2}} ⋊_{α} ℤ_{2}.

If θ_{1}, θ_{2} ∈ (0, 1) with θ_{1} = ±θ_{2} mod ℤ, then θ_{2} = θ_{1} or θ_{2} = 1 − θ_{1}. Thus, it remains to prove that A_{θ} ⋊_{α} ℤ_{2} ≅ A_{1−θ} ⋊_{α} ℤ_{2} for θ ∈ (0, 1) ∖ ℚ. By Theorem 6.3 and George Elliott’s classification theorem for AF algebras (Theorem 4.3 of [23]), it suffices to exhibit an isomorphism f: K_{0}(A_{θ} ⋊_{α} ℤ_{2}) → K_{0}(A_{1−θ} ⋊_{α} ℤ_{2}) of scaled ordered K_{0}-groups. Since each algebra has a unique tracial state, and since the order on the K_{0}-group of a simple unital AF algebra is determined by its tracial state (this is true much more generally: see Corollary 5.7 and Theorems 5.8 and 6.8 of [40]), it suffices to find a group isomorphism f as above such that (τ_{2, 1−θ})_{∗} ∘ f = (τ_{2,θ})_{∗} and f([1]) = [1].

Define f on the basis elements listed in Theorem 4.9(1) by f([E_{θ_{1}}^{ℤ_{2}}]) = [p^{θ}] − [E_{θ_{2}}^{ℤ_{2}}], and by sending each of the other basis elements for K_{0}(A_{θ} ⋊_{α} ℤ_{2}) to the corresponding basis element for K_{0}(A_{1−θ} ⋊_{α} ℤ_{2}), that is, f([1]) = [1], f([p^{θ}]) = [p^{1−θ}], etc. The computations of the trace on the generators given in Theorem 4.9(1) show that (τ_{2, 1−θ})_{∗} ∘ f = (τ_{2,θ})_{∗}, as desired. □

In the cases F = ℤ_{2} and F = ℤ_{4}, it also is easy to use the isomorphism φ: A_{θ} → A_{1−θ}, given by u_{θ} ↦ v_{1−θ} and v_{θ} ↦ u_{1−θ}, to directly exhibit an isomorphism A_{θ} ⋊_{α} F ≅ A_{1−θ} ⋊_{α} F. Presumably this can be done in the other two cases as well.

Theorem 6.3 also has the following corollary.

Corollary 6.5. Under the hypotheses of Theorem 6.3, the fixed point algebra A_{θ}^{F} is AF.

Proof. By the Proposition in [66], the fixed point algebra A_{θ}^{F} is isomorphic to a corner of A_{θ} ⋊_{α} F. □

We now turn to the crossed product of a simple higher dimensional noncommutative torus by the flip, defined before Corollary 5.12. The following theorem generalizes Theorem 3.1 of [6], and completely answers a question raised in the introduction to [28].

Theorem 6.6. Let Θ be a nondegenerate real d × d skew symmetric matrix. Let φ: ℤ_{2} → Aut(A_{Θ}) be the flip action. Then A_{Θ} ⋊_{φ} ℤ_{2} is an AF algebra.

Proof. By Theorem 3.5 of [60], we know that A_{Θ} is a simple separable unital C*- algebra with tracial rank zero. By Corollary 5.12, we may apply Corollary 1.6 and Theorem 2.6 of [59] to conclude that A_{Θ} ⋊_{φ} ℤ_{2} is a simple separable unital C*-algebra with tracial rank zero.

Lemma 2.1 implies that the cocycle ω_{Θ} extends to a cocycle ω̃_{Θ} on the semidirect product ℤ^{d} ⋊ ℤ_{2} such that C^{∗}(ℤ^{d} ⋊ ℤ_{2}, ω̃_{Θ}) ≅ A_{Θ} ⋊_{φ} ℤ_{2}. Since ℤ^{d} ⋊ ℤ_{2} can be embedded into ℝ^{d} ⋊ ℤ_{2}, Corollary 6.2 therefore implies that A_{Θ} ⋊_{φ} ℤ_{2} satisfies the Universal Coefficient Theorem.

Since ω_{Θ} and ω̃_{Θ} are real cocycles (Definition 1.12), it follows from Corollary 1.13 that

K_{0}(A_{Θ} ⋊_{φ} ℤ_{2}) = K_{0}(C^{∗}(ℤ^{d} ⋊ ℤ_{2}, ω̃_{Θ})) ≅ K_{0}(C^{∗}(ℤ^{d} ⋊ ℤ_{2})),

and K_{1}(A_{Θ} ⋊_{φ} ℤ_{2}) ≅ K_{1}(C^{∗}(ℤ^{d} ⋊ ℤ_{2})) for all Θ. But in [28] it was shown that K_{0}(A_{Θ} ⋊_{φ} ℤ_{2}) ≅ ℤ^{3·2^{d−1}} and K_{1}(A_{Θ} ⋊_{φ} ℤ_{2}) = 0 for some special values for Θ. This must then be true for all Θ. Therefore Lin’s classification theorem (Theorem 5.2 of [41]), in the form presented in Proposition 3.7 of [60], implies the conclusion. □

Corollary 6.7. Let Θ be a nondegenerate real d × d skew symmetric matrix. Let φ: ℤ_{2} → Aut(A_{Θ}) be the flip action. Then the fixed point algebra A_{Θ}^{ℤ_{2}} is AF.

Proof. The proof is the same as for Corollary 6.5. □

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Westfälische Wilhelms-Universität Münster, Mathematisches Institut, Einsteinstr. 62, D-48149 Münster, Germany

E-mail address: echters@math.uni-muenster.de

Westfälische Wilhelms-Universität Münster, Mathematisches Institut, Einsteinstr. 62, D-48149 Münster, Germany

E-mail address: lueck@math.uni-muenster.de

Department of Mathematics, University of Oregon, Eugene OR 97403-1222, USA.

E-mail address: ncp@darkwing.uoregon.edu

Department of Mathematics and Computer Science, The University of Northern British Columbia, Prince George, B.C. V2N 4Z9, Canada

E-mail address: walters@hilbert.unbc.ca or walters@unbc.ca